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QUESTION

# Find the local maximum and minimum values of f using both the First and Second Derivative Tests? f(x) = x / (x^2 + 25)

 f(5)=1/10 is a local max.  f(-5)=-1/10 is a local min.

f(x)={x}/{x^2+25}

By ,

f'(x)={1 cdot(x^2+25)-x cdot 2x}/{(x^2+25)^2}={25-x^2}/{(x^2+25)^2}=0

=> x= pm5" " (Critical Numbers)

First Derivative Test

{(f':(-) to (+)" at "x=-5 => f(-5)=-1/10" is a local min."),(f':(+) to (-)" at "x=5 => f(5)=1/10" is a local max."):}

Second Derivative Test

By Quotient Rule,

f''(x)={ -2x(x^2+25)^2-(25-x^2)cdot2(x^2+25)(2x)}/{(x^2+25)^4}={2x(x^2-75)}/(x^2+25)^3

{(f''(-5)>0 => f(-5)=-1/10" is a local min."),(f''(5)<0 => f(5)=1/10" is a local max."):}

I hope that this was helpful.