Question

(a) If $a$ is a nonzero constant, find all critical points of \[ f(x)=\frac{a}{x^{2}}+x \] (b) Use the second-derivative test to show that if $a$ is positive then the graph has a local minimum, and if $a$ is negative then the graph has a local maximum.

          (a) If $a$ is a nonzero constant, find all critical points of
\[
f(x)=\frac{a}{x^{2}}+x
\]
(b) Use the second-derivative test to show that if $a$ is positive then the graph has a local minimum, and if $a$ is negative then the graph has a local maximum.

Added by Mark S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

06/23/2023

Step 1

First, we need to find the first derivative of the function $f(x)$: $f'(x) = \frac{d}{dx} \left(\frac{a}{x^2} + x\right)$ Using the power rule and the sum rule for differentiation, we get: $f'(x) = -\frac{2a}{x^3} + 1$ Now, we need to find the critical points Show more…

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(a) If $a$ is a nonzero constant, find all critical points of \[ f(x)=\frac{a}{x^{2}}+x \] (b) Use the second-derivative test to show that if $a$ is positive then the graph has a local minimum, and if $a$ is negative then the graph has a local maximum.