What are the local extrema, if any, of $f(x)= 2x+15x^(2/15)$ ?

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Answer 1 · 2017-07-23T13:36:05

Local maximum of 13 at 1 and local minimum of 0 at 0.

Domain of $f$ is $RR$

$f'(x) = 2+2x^(-13/15) = (2x^(13/15)+2)/x^(13/15)$

$f'(x) = 0$ at $x = -1$ and $f'(x)$ does not exist at $x = 0$ .

Both $-1$ and $9$ are in the domain of $f$ , so they are both critical numbers.

First Derivative Test:

On $(-oo,-1)$ , $f'(x) > 0$ (for example at $x = -2^15$ )
On $(-1,0)$ , $f'(x) < 0$ (for example at $x = -1/2^15$ )

Therefore $f(-1) = 13$ is a local maximum.

On $(0,oo)$ , $f'(x) >0$ (use any large positive $x$ )

So $f(0) = 0$ is a local minimum.

What are the local extrema, if any, of f(x)= 2x+15x^(2/15)f(x)= 2x+15x^(2/15)?