Question

Find the x-coordinate where the graph of the function f(x) = e(–cosx) has a slope of 0.

Answer 1

The slope of the tangent line to $f(x)$ through the point $(x,f(x))$ is given by the derivative:

$f(x)=e^{-\cos x}\implies f'(x)=e^{-\cos x}\sin x$

The value of the derivative is zero when $\sin x=0$, which happens for infinitely many points $x=n\pi$ where $n$ is any integer.