Usage of Maximum Modulus Theorem?

Question

I have a function $f(z)=ze^z$, I want to find the maximum value of $|f(z)|$ as $z$ varies over the region $D=\{x+iy: x^2+y^2\leq 4, x,y\geq 0\}$.

I was thinking that this is case where I can use the maximum modulus principle, because $f(z)$ is analytic and nonconstant ton the domain $D$, MMP stated that $f$ has no local max. Therefore $ze^z$ has no local max as $z$ varies over $D$. Trick quesion?

Answer 1

$$|f(z)|=|z||e^z|=|z|e^x\leq2e^2$$ so, the maximum value of $|f(z)|$ is $2e^2$

Answer 2

since the function is analytic at the region, you will have maximum or minimum on the boundary i.e. at the path $z = 2 e^{i \theta}$