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 Tiffany C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

06/23/2023

Step 1

To do this, we find the first derivative of the function and set it equal to zero. $$ f'(x) = -\frac{2a}{x^3} + 1 $$ Now, we set $f'(x)$ equal to zero and solve for $x$: $$ -\frac{2a}{x^3} + 1 = 0 $$ $$ \frac{2a}{x^3} = 1 $$ $$ x^3 = 2a $$ Show more…

Show all steps

Thanks for your feedback!

(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.