Find all absolute and relative extreme values of the function f(x) = -x + 2x< - x – 1,on the interval (-2, 2].

Answered: Find all absolute and relative extreme…

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter11: Rational And Irrational Numbers

Section: Chapter Questions

Problem 23CLR

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Question

Find all absolute and relative extreme values of the function f(x) = -x + 2x< - x – 1,
on the interval (-2, 2].

Expert Solution

Step 1

The given function is $f (x) = - x^{3} + 2 x^{2} - x - 1$ on $[- 2, 2]$ .

Compute the critical points as follows.

$f' (x) = 0 \frac{d}{d x} (- x^{3} + 2 x^{2} - x - 1) = 0 - 3 x^{2} + 4 x - 1 = 0 - 3 x^{2} + 3 x + x - 1 = 0 - 3 x (x - 1) + (x - 1) = 0 (x - 1) (- 3 x + 1) = 0 x = 1, \frac{1}{3}$

Thus, the critical points are $x = 1, \frac{1}{3}$ .

Step 2

Obtain the second derivative as follows.

$\begin{array}{rcl} f'' (x) & = & \frac{d}{d x} (- 3 x^{2} + 4 x - 1) \\ = & - 6 x + 4 \end{array}$

Note that at $x = 1$ , $f'' (x) < 0$ . The relative maximum exist at $x = 1$ .

at $x = \frac{1}{3}$ , $f'' (x) > 0$ . The relative minimum exist at $x = \frac{1}{3}$ .

Compute the function value at critical values and end points of $[- 2, 2]$ as follows.

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