Question

Question #801ba

Answer 1

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

Explanation:

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

By Quotient Rule,

`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)

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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."):}`

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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."):}`

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I hope that this was helpful.