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Chapter 3: Q. 3 (page 286)

When you try to find the local extrema of a function f on an interval I, one of the first steps is to find the critical points of f . Explain why these critical points of f won鈥檛 help you locate any 鈥渆ndpoint鈥� extrema.

Short Answer

Expert verified

The critical points of the function won't help to locate any endpoints extrema because the derivative of the function may or may not be zero or fails to exist at the endpoints of the interval. The critical point is always calculated by equating the first derivative of the function equal to zero.

Step by step solution

01

Step 1. Given information.

We have to explain that when we try to find the local extrema of a function f on an interval I, one of the first steps is to find the critical points of f ,why these critical points of f won鈥檛 help us locate any 鈥渆ndpoint鈥� extrema.

02

Step 2. Explanation

The critical points of the function won't help to locate any endpoints extrema because the derivative of the function may or may not be zero or fails to exist at the endpoints of the interval. The critical point is always calculated by equating the first derivative of the function equal to zero.

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Most popular questions from this chapter

Sketch careful, labeled graphs of each function f in Exercises 63鈥�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',andf'', and examine any relevant limits so that you can describe all key points and behaviors of f.

f(x)=e3x-e2x

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.

For each graph of f in Exercises 49鈥�52, explain why f satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b] and approximate any values c 鈭� (a, b) that satisfy the conclusion of the Mean Value Theorem.

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

fx=3x-2x

Find the critical points of the function

f'(x)=(x-1)(x-2)x-3
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