Please help with a step by step solution to this problem 4. Find all local maximums and local minimums for $ f(x) = x^3 - 2x^2 + x - 1 $.

Answered: 4. Find all local maximums and local…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter2: Functions

Section: Chapter Questions

Problem 8CC: What does it mean to say that f(4) is a local maximum valueof f ?

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Please help with a step by step solution to this problem

4. Find all local maximums and local minimums for $ f(x) = x^3 - 2x^2 + x - 1 $.

Expert Solution

Step 1

We have given function

$f (x) = x^{3} - 2 x^{2}$ $+ x - 1$

Derivative of the function is given as

$f^{'} (x) =$ $\frac{d}{d x} (f (x))$

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

$f^{'} (x) = 3 x^{2} - 2$ $(2 x) + 1 - 0$

$f^{'} (x) = 3 x^{2} -$ $4 x + 1$ ......... (i)

To get critical points put $f^{'} (x) = 0$

$3 x^{2} - 4 x + 1$ $= 0$

$3 x^{2} - 3 x - x + 1$ $= 0$

$3 x (x - 1) - 1$ $(x - 1) = 0$

$(3 x - 1) (x - 1) = 0$

$3 x - 1 = 0$ or $x - 1 = 0$

$x = \frac{1}{3}$ or $x = 1$

$x = \frac{1}{3}, 1$

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4. Find all local maximums and local minimums for f(x) = x³ - 2x² + x - 1.