I must tell the veracity of the affirmation : $\displaystyle f(x,y)=\sin (x) + \cos (y) +a(x+y) $ has a unique minimum if $\displaystyle a>1$.

My attempt : $\displaystyle \frac{\partial f}{\partial x}(x,y)=\cos (x) +a$.

$\displaystyle \cos (x) +a=0 \Leftrightarrow \cos (x)=-a$ which never happens since $\displaystyle a>1$. Therefore $\displaystyle f$ doesn't have a minimum, so the affirmation is false.

Am I right?