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Essential Calculus: Early Transcendentals
Found in: Page 208
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Explain the difference between an absolute minimum and a local minimum.

The local minimum of \(f(x)\) occurs at \(x = 3\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Given data

The given function is absolute minimum and local minimum.

Step 2: Concept of Differentiation

Differentiation is a method of finding the derivative of a function. Differentiation is a process, where we find the instantaneous rate of change in function based on one of its variables.

Step 3: Simplify the expression

Absolute minimum: A function \(f\) has an absolute minimum at \(c\) if \(f(c) \le f(x)\quad \forall x\) in the domain \((D)\) of \(f\).

Local minimum: A function \(f\) has local minimum at \(c\) if there is an open interval L (containing \(c\) any \(x\) near to \(c\)) such that \(f(c) \le f(x)\quad \forall x\) in L. Local minimum and local maximum do not attain at extreme values.

Difference between the absolute minimum and the local minimum: From the above, the local minimum is any one of the minimum value available in the curve whereas the absolute minimum is the highest minimum of the curve. That is, the least point of the curve is called as absolute minimum. Therefore, it can be concluded that any absolute minimum is called as local minimum but not the vice versa.

Step 4: Sketch the graph of the example

Consider an example as shown below in Figure:

From Figure, it is observed that, \(f(1) \le f(x)\quad \forall x\) in \(R\).

Thus, the absolute minimum of \(f(x)\) occurs at \(x = 1\).

Also, it can be observed that, \(f(3) \le f(x)\quad \forall x\) nearer to 3.

Thus, the local minimum of \(f(x)\) occurs at \(x = 3\).

Most popular questions for Math Textbooks

Sketch the graph of a function that is continuous on (1, 5) and has the given properties.

8. Absolute minimum at 1, absolute maximum at 5, local maximum at 2, local minimum at 4.

(a) Show that \({e^x} \ge 1 + x\) for \(x \ge 0\).

(b) Deduce that \({e^x} \ge 1 + x + \frac{1}{2}{x^2}\) for \(x \ge 0\).

(c) Use mathematical induction to prove that for \(x \ge 0\) and any positive integer \(n\),

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Show that the inflection points of the curve \(y = x\sin x\) lie on the curve \({y^2}\left( {{x^2} + 4} \right) = 4{x^2}\).

Use the graph to state the absolute and local maximum and minimum values of the function.

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