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July 13th, 2017, 01:13 AM   #1
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Local extrema

I'm given the function f(x,y)=(1+e^y)sinx-ye^y and I need to prove that this function has Infinitely many max and none min. I know how to find those, but I'm stuck with this Infinitely many. Thank you
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July 13th, 2017, 01:41 PM   #2
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I guess the infinitely many comes from the fact that you have a periodic function, i.e, the sine. When you calculate the gradient of this function, calculate the second derivative and proof that is always negative for the maximum points, which should be distributed with integer numbers
\begin{eqnarray}
\sin(x)=0 && x=2\pi n
\end{eqnarray}
where $n$ is any positive or negative integer. Since you have a sine, the derivative will give you a cos, so its all about to find the points where a cos is zero:
\begin{eqnarray}
\cos(x)=0 && x=(2n+1)(\pi/2)
\end{eqnarray}

hope it helps now!
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