Find the critical points. f(x) = x2 log,(x² + 2)Find the critical numbers of the function.

fx=x2log7x2+2⇒fx=x2logx2+2ln7 ∵logab=logblna Critical numbers are the values of x for which…

Answered: Find the critical numbers of the…

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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Question

Find the critical points.

f(x) = x2 log,(x² + 2)
Find the critical numbers of the function.

Expert Solution

Given,

$f (x) = x^{2} \log_{7} (x^{2} + 2) \Rightarrow f (x) = \frac{x^{2} \log (x^{2} + 2)}{\ln 7} [∵ \log_{a} b = \frac{\log b}{\ln a}]$

Critical numbers are the values of x for which first derivative is zero.

So first we find the derivative of given function.

$\begin{array}{rcl} f^{/} (x) & = & \frac{1}{\ln 7} [\frac{d (x^{2})}{d x} \cdot \log (x^{2} + 2) + \frac{d (\log (x^{2} + 2))}{d x} \cdot x^{2}] \\ = & \frac{1}{\ln 7} [2 x \cdot \log (x^{2} + 2) + \frac{1}{x^{2} + 2} \cdot \frac{d (x^{2} + 2)}{d x} \cdot x^{2}] \\ = & \frac{1}{\ln 7} [2 x \cdot \log (x^{2} + 2) + \frac{1}{x^{2} + 2} \cdot 2 x \cdot x^{2}] \\ = & \frac{2 x}{\ln 7} [\log (x^{2} + 2) + \frac{x^{2}}{x^{2} + 2}] \\ = & \frac{2 x}{\ln 7} [\frac{(x^{2} + 2) \log (x^{2} + 2) + x^{2}}{x^{2} + 2}] \end{array}$