Answer
The function is odd.
Zeros: $x=0, \pi k + \frac{\pi}{2}$ where k is an integer
Work Step by Step
To check whether a function f is even, odd, or neither, evaluate $f(-x)$. If $f(-x)=f(x)$ then the function is even. If $f(-x)=-f(x)$ then the function is odd. Otherwise the function is neither:
$f(x)=x\cos x$
$f(-x)=-x\cos(-x)$
$f(-x)=-x\cos x$
$f(-x)=-f(x)$
Therefore the function is odd.
The zeros of the function occur whenever $x=0$ or $\cos x = 0$
So the zeros are $x=0, \pi k + \frac{\pi}{2}$ where k is an integer
