The greatest value of $\cos(xe^{\lfloor x \rfloor} +7x^2 -3x)$

Question

The greatest value of $g(x)=\cos(xe^{\lfloor x \rfloor} +7x^2 -3x)$ , $x\in [-1,\infty)$ , is

My work:

For function to be maximum $f(x) = xe^{\lfloor x \rfloor} +7x^2 -3x$ must be minimum

When $ x $ is integer $$f(x)=xe^{x} +7x^2 -3x$$ $$f'(x)=e^x +x.e^x +14x -3 =0 $$ $$ =e^x(1+x) +(14x -3)=0$$

But from here I am not able to find the $ x $ and what should I do when $ x$ is not integer

Answer 1

Cosine can never become larger than $1$. In this case it can actually become $1$.

$g(x)$ has maximum not for minima of $f$, but wherever $f(x)=2\pi n$ for some integer $n$.