Maximum Modulus of Complex Function

Question

here's a question I'm working on that I'm a bit stuck on.
Let:
$f(z) = \frac{z^2}{z + 2}$
Find the maximum value of $|f(z)|$ as $z$ varies over the unit disc.

Since $f(z)$ is analytic $\forall z$ in the region,and $f(z)$ is non-constant, the maximum of this function will be on the boundary of the unit disc (Is this the correct application of the theorem?)
If we let $z = e^{it}$ then
$|f(e^{it})| = \frac{|e^{it}|^2}{|e^{it} + 2|}$
$|f(e^{it})| = \frac{1}{|e^{it} + 2|}$

However, I'm stuck here.
The answer given is $1$, but I'm not quite sure how to figure that out. There's quite a bit of confusion on my end as well, as I'm not fully certain if I'm on the right track.

Any help would be appreciated.
Thanks.

Answer 1

The minimum of $\mid e^{it}+2\mid$ pretty clearly occurs when $t=\pi$. That is, the maximum modulus is $1$.

You did the problem correctly (up to there).

Answer 2

Hint: do this by common sense: you want to maximize $\frac{1}{|e^{it} + 2|}$, which means minimizing $|e^{it} + 2|$, so draw the picture, and find a complex number on the unit circle whose distance from $z=-2$ is minimum.