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…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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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.