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Q. 78

Expert-verified
Calculus
Found in: Page 276
Calculus

Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Illustration

Short Answer

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f, f', and f'', and examine any relevant limits so that you can describe all key points and behaviors of f.

f(x)=e3x-e2x

The sketch of the graph is

The sign chart is

See the step by step solution

Step by Step Solution

Step 1. Given Information.  

The given function is f(x)=e3x-e2x.

Step 2. Finding the roots.  

To find the roots we will put the given function equal to zero.

So,

f(x)=e3x-e2x0=e3x-e2xx=0

Therefore, the given function has roots at x=0.

Step 3. Testing the signs.   

Now, let's test the sign for f' and f''.

Let's differentiate the equation to find f'.

So,

f'(x)=3ex-2e2x0=3ex-2e2x0=3ex-22=3ex23=exx=ln23

Thus, f' has a local minimum at x=ln23. It is positive on the interval 0, and negative elsewhere. Hence the graph of f will be increasing during the positive interval and decrease during the negative interval.

Let's differentiate again.

So,

f''(x)=9ex-4e2x

Thus, f'' is positive on the interval 0, and negative elsewhere. Hence, the graph of f will be concave up on positive interval and concave down elsewhere.

Step 4. Sketch the sign chart. 

The sign chart is

Step 5. Examine the relevant limit. 

Let's examine the limits.

limxf(x)=limx-f(x)=0

Step 6. Sketch the graph of function f.  

The graph of the function is

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