I have a function $F(z_1,z_2) := z_1^2\cos^2(z_2)+\frac{3}{4}z_2^2$.
I need to show that $\forall \alpha > 0\, \exists \gamma > 0\, \forall z\, \|z\| \geq \alpha \implies F(z_1,z_2) \geq \gamma $.
Aside from using Lagrange multipliers for this function $F$ with restriction $z_1^2 + z_2^2 \geq \alpha^2$ that give a basically unsolvable (?) equation over $z_2$:
$-(\alpha^2 - z_2^2)\sin(2z_2) + \frac{3}{2}z_2 -z_2\cos^2(z_2) = 0$ do you guys have any other ideas I could use here?
I'm asking this because this is an example taken from a book so I feel like it should not be that hard, maybe so…