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Derivatives and the Shape of a Graph

Derivatives and the Shape of a Graph

Hiking can be a way of connecting calculus with nature. Walking up a hill feels completely different from walking it down, and in most instances, it is easier to go up than go down. Not to mention that some hills can be steeper than others!

By just taking a look at a hill you can know beforehand if it will be easy to climb or not, its shape is actually giving you information!

Rather than thinking of a hill, you can picture a graph instead. The shape and the derivatives of a function can give you some crucial information about its behavior, so here we will look at how Derivatives and the Shape of a Graph are related.

Derivatives and the Shape of a Graph Meaning

Talking about the shape of a graph might sound a little vague at first. What kind of shapes are present in a graph?

In Calculus, the shape of a graph refers to important features of different sections of the graph:

  • Is it increasing or decreasing?
  • Is it above or below the x-axis?
  • Does it have a steep slope?
  • Its tangent lines lie below or above the graph?

The above features might apply at different sections, or intervals, of the graph. A graph can be increasing in one interval and then decrease in another!

Relationship between Derivatives and the Shape of a Graph

Now that you have seen the meaning of the shape of a graph in Calculus, you might be wondering how derivatives are involved.

Derivatives measure change, so having the derivative of a function is key to knowing how its graph is changing.

The first derivative tells you whether the graph is increasing or decreasing.

The second derivative tells you whether the graph curves up or curves down.

Knowing this information is enough to roughly sketch the graph of a function without using graphing software!

The First Derivative Effects on the Shape of a Graph

As stated before, the first derivative of a function tells you if a function is increasing or decreasing in a certain interval. Here is how it works:

  • A function \(f (x) \) is increasing in an interval where its derivative is positive, that is \( f'(x) > 0\).
  • A function is decreasing in an interval where its derivative is negative, that is \( f'(x) <0\).

You can also give a graphical interpretation to the above statements, that is

  • A function is increasing in an interval where the slope of a line tangent to its graph is positive.
  • A function is decreasing in an interval where the slope of a line tangent to its graph is negative.

Derivatives and the Shape of a Graph graph of a parabola with two tangent lines one where the function is decreasing and another where the function is increasing StudySmarterFigure 1. Decreasing and increasing intervals of a function

The points where \( f'(x)=0 \) are known as critical points. In functions with continuous derivatives, critical points are likely to be points where a function switches from increasing to decreasing or vice-versa. For more information about this topic please take a look at our article about the First Derivative Test!

The Second Derivative Effects on the Shape of a Graph

The second derivative of a function \( f(x) \) is denoted as \( f''(x)\), and it can be found by differentiating the first derivative of the function, that is, by differentiating a function twice in a row. The second derivative of a function, assuming it exists, tells you which way the function bends. There are two special words in Calculus for this idea: Concave and convex.

A function is said to be concave down, or just concave, in an interval where its second derivative is negative. The lines tangent to the function's graph inside an interval where it is concave will lie above the graph.

Derivatives and the Shape of a Graph a line tangent to a concave parabola is above the graph of the parabola StudySmarterFigure 2. A line tangent to a function in an interval where it is convex lies above the graph

And what if the tangent lines lie below the graph?

A function is said to be concave up, or convex, in an interval where its second derivative is positive. The lines tangent to the function's graph inside an interval where it is convex will lie below the graph.

Derivatives and the Shape of a Graph a line tangent to a convex parabola is below the graph of the parabola StudySmarterFigure 3. A line tangent to a function in an interval where it is convex lies below the graph

The concavity of a graph is independent of whether it is increasing or decreasing! You can have, for instance, a concave decreasing interval or a convex decreasing interval. All four combinations are possible!

In the following graphs, you can look at the difference between concave and convex functions with graphs.

Both functions below are increasing. However, notice how they bend differently.

Derivatives and the Shape of a Graph a line tangent to a function that is both increasing and concave StudySmarterFigure 4. The tangent line is above the function, so it is concave

Derivatives and the Shape of a Graph a line tangent to a function that is both increasing and convex StudySmarterFigure 5. The tangent line is below the function, so it is convex

Now both functions below are decreasing. Pay close attention to the bending.

Derivatives and the Shape of a Graph a line tangent to a function that is both decreasing and concave StudySmarterFigure 6. The tangent line is above the function, so it is concave

Derivatives and the Shape of a Graph a line that is tangent to a function that is both decreasing and convex StudySmarterFigure 7. The tangent line is below the function, so it is convex

The points where a function switches from concave to convex, or vice-versa, are called inflection points. For more information about this topic, please reach out to our article about the Second Derivative Test.

Examples of first and second derivative effects on the shape of a graph

The first derivative of a function can be used to find intervals where a function is increasing or decreasing. Here is an example of how this is done.

Determine the intervals where the function

\[f(x)=\frac{1}{3}x^3-4x+1\]

is increasing and/or decreasing.

Solution:

Since you need to find the intervals where the given function is increasing and/or decreasing, you should begin by finding its derivative. You can achieve this with the Power Rule, that is

\[ f'(x) = x^2-4.\]

To find where the function is increasing, you need to solve the inequality

\[f'(x)>0,\]

that is

\[x^2-4>0,\]

which you can factor as

\[(x+2)(x-2)>0.\]

The above inequality states that the product of two expressions is greater than zero. This means that both expressions have the same sign, so either

\[ x+2>0 \quad \text{and} \quad x-2>0 \]

or

\[ x+2<0 \quad \text{and} \quad x-2<0.\]

Solving the above compound inequality tells you that \( x>2 \) or \(x<-2\), so the function is increasing in the interval \( (-\infty,-2) \) and in the interval \( (2,\infty)\).

To find where the function is decreasing, you can solve the inequality

\[ f'(x) <0,\]

but since you already solved an equality like this, the remaining interval is the decreasing interval, so the function is decreasing in the interval \( (-2,2) \).

To see if your result makes sense, you should finish by taking a look at the graph of the given function.

Derivatives and the Shape of a Graph graph of the cubic function with vertical lines splitting each interval where there are two increasing intervals and one decreasing interval StudySmarterFigure 8. The function has one decreasing interval and two increasing intervals

You can also find the concavity of the given function by using its second derivative.

Determine the intervals where the function

\[f(x)=\frac{1}{3}x^3-4x+1\]

is concave and/or convex.

Solution:

Previously, you found the derivative of the given function using the Power Rule, that is

\[ f'(x)= x^2-4.\]

By using the Power Rule again, you can find the second derivative, so

\[ f''(x)=2x.\]

To find the intervals where the function is concave you need to solve the inequality

\[ f''(x) < 0,\]

that is

\[2x <0,\]

whose solution is

\[ x<0.\]

This means that the function is concave in the interval \( (-\infty,0) \).

To find the intervals where the function is convex, you need to solve the inequality

\[ f''(x) >0, \]

but since you just solved a similar inequality you can just flip the inequality sign, so

\[ x>0\]

gives you the interval where the function is convex, that is \( (0,\infty)\).

Derivatives and the Shape of a Graph  graph of the cubic function with a vertical line splitting a concave interval from a convex interval StudySmarterFigure 9. The function is concave in one interval, and convex in another

Derivatives and the Shape of a Graph - Key takeaways

  • The derivative of a function can tell whether a function is increasing or decreasing in an interval.
    • If \( f'(x) >0\), \( f(x) \) is increasing.
    • If \( f'(x) <0\), \( f(x) \) is decreasing.
  • Depending on how it bends, a function can be concave down (or just concave), or it can be concave up (or convex).
  • The second derivative of a function can tell whether a function is concave or convex.
    • If \( f''(x) <0\), \( f(x) \) is concave.
    • If \( f''(x) >0\), \( f(x) \) is convex.

Frequently Asked Questions about Derivatives and the Shape of a Graph

The first derivative tells whether the function is increasing or decreasing. The second derivative tells whether the function is concave upward or concave downward.

Evaluating the derivatives can help us to sketch an accurate graph of a function without using graphing software.

The first derivative shows whether the function is increasing or decreasing.

The second derivative shows whether the function is concave upward or concave downward.

The first derivative tells whether the function is increasing or decreasing. The second derivative tells whether the function is concave upward or concave downward.

Final Derivatives and the Shape of a Graph Quiz

Question

What is a critical point?

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Answer

A critical point is an x-value for which the derivative of a function is equal to 0.

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Question

A critical point is also known as a:

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Answer

Stationary Point

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Question

Every critical point is either a local maximum or a local minimum.

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Answer

False.

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Question

Every local extremum is a critical point.

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Answer

False.

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Question

The slope of a line tangent to a function at a critical point is equal to zero.

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Answer

True.

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Question

A function can have more than one critical point.

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Answer

True.

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Question

Does the absolute value function have any critical points?

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Answer

No.

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Question

Let \( c \) be a critical point of a function \( f(x). \) What does The Second Derivative Test tells us if \( f''(c) <0 \)?

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Answer

The critical point is a local maximum.

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Question

Let \( c \) be a critical point of a function \( f. \) What does The Second Derivative Test tells us if \( f''(c) >0 \)?

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Answer

The critical point is a local minimum.

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Question

Let \( c \) be a critical point of a function \( f. \) What does The Second Derivative Test tells us if \( f''(c)=0 \)?


Show answer

Answer

The test is inconclusive.

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Question

How do you find the critical points of a function?

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Answer

The critical points of a function can be found by doing The First Derivative Test.

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Question

If The Second Derivative Test becomes inconclusive then a critical point is neither a local maximum or a local minimum.

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Answer

False.

Show question

Question

A function is said to be concave down, or concave, in an interval where:

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Answer

Its second derivative is negative.

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Question

A function is said to be concave up, or convex, in an interval where:

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Answer

Its second derivative is positive.

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Question

What is an inflection point?

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Answer

An x-value for which the concavity of a graph changes.

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Question

A critical point of the function \( g(x)= 2x^3+x^2-1\) is \( x=0. \) Its second derivative is \( g''(x)=12x+2.\) Is the critical point a relative maximum or a relative minimum?

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Answer

Relative minimum

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Question

The second derivative of a function is \( f''(x)=12x^2-2. \) Is the function concave or convex at \(x=1\)?

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Answer

Convex.

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Question

The function \( h(x)= x^2+1 \) has a critical point at \( x=0. \) Is this a relative maximum or a relative minimum?

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Answer

Relative minimum.

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Question

The second derivative of a function is \( g''(x)= -2x.\) Is it concave or convex at \( x=2 \)?

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Answer

Concave.

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Question

What are the conditions that a function needs to meet in order to guarantee that The Candidates Test works?

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Answer

The function and its derivative need to be continuous and defined over a closed interval.

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Question

Every local maximum is also a global maximum.

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Answer

False.

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Question

The global maximum of a function is always a critical point.

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Answer

False.

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Question

The Candidates Test can be used if the function is continuous, defined over a closed interval, but not differentiable.

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Answer

False.

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Question

The Candidates Test can be used if the function is continuous, differentiable, but defined over an open interval.

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Answer

No.

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Question

A function can have more than one local minimum.

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Answer

True.

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Question

A function can have more than one global maximum.

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Answer

False.

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Question

The first derivative tells whether the graph of a function is ____.

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Answer

increasing or decreasing.

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Question

The second derivative tells whether the graph of a function is ____.

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Answer

concave up or concave down

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Question

If \( f'(x) > 0\), then \( f(x) \) is ____.

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Answer

increasing.

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Question

If \( f'(x) <0 \), then \( f(x) \) is ____.

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Answer

decreasing.

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Question

If \( f''(x) > 0\), then \( f(x) \) is ____.

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Answer

convex.

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Question

If \( f''(x) < 0 \), then \( f(x) \) is ____.

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Answer

concave.

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Question

Every function has a critical point.

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Answer

False.

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Question

A line tangent to a curve at a critical point is a(n) _______ line.

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Answer

horizontal

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Question

The second derivative of a function is \( r''(x)=x^2-3x \). Is it concave or convex at \(x=3\) ?

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Answer

\( x=3 \) is an inflection point

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Question

The Candidates Test can be used even if a function has no critical points.

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Answer

True.

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Question

Why is it necessary to narrow down the search of the global extrema of a function?

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Answer

Because most functions will have an infinite amount of outputs, so sorting them is practically impossible.

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Question

What are the steps to performing a Candidates Test?

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Answer

  1. Find all the critical points of the function.
  2. Evaluate the function at all of its critical points.
  3. Evaluate the function at both ends of its domain.
  4. Sort all of the above values. 

Show question

Question

True/ False: A function can have more than one increasing interval.

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Answer

True.

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Question

True/False: Convex is the same as concave down.

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Answer

False.

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Question

True/False: An increasing function can be convex as well.

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Answer

True.

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Question

True/False: An increasing function can be concave as well.

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Answer

True. 

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Question

True/False: A parabola is either completely concave or completely convex.

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Answer

True.

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Question

True/False: A straight line is always represented by an increasing function.

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Answer

False.

Show question

Question

True/False: A function that is represented by a non-horizontal straight line is either an increasing or a decreasing function.


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Answer

True.

Show question

Question

The point where a function switches from concave to convex, or vice-versa, is called a(n) ____ point.

Show answer

Answer

inflection.

Show question

Question

Consider the function
\[ f(x)= 5x+1.\]
Is this an increasing, decreasing, or constant function?

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Answer

Increasing.

Show question

Question

Consider the function
\[ f(x)=4-x.\]
Is this an increasing, decreasing, or constant function?

Show answer

Answer

Decreasing.

Show question

Question

Consider the function
\[ g(x)= \frac{\pi}{4}+1.\]
Is this an increasing, decreasing, or constant function?

Show answer

Answer

Constant.

Show question

Question

True/False: A linear function is neither concave nor convex.

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Answer

True.

Show question

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