I need a small hint with convex functions.
Find all $a,b,c$ such that $f(x,y,z)=x^2 + 2axy+by^2+cz^2$ is convex, strictly convex and strongly convex.
To answer if a function is convex and strictly convex it is enough to find all $a,b,c$ such that it's Hessian matrix is positive-defined, e.g $\nabla^2f \succ 0$ ($\nabla^2f \succeq 0 $).
So,
$$ \nabla^2f= \begin{bmatrix} 2 & 2a & 0 \\ 2a & 2b & 0 \\ 0 & 0 & 2c \end{bmatrix} \succ 0 $$
According to Sylvester's criterion, we get
$$ \begin{cases} b-a^2 > 0 \\ c(b-a^2) > 0 \end{cases} $$
Which means that
$$ \begin{cases} -\sqrt{b} < a < \sqrt{b} \\ b > 0 \\ c > 0 \end{cases} $$
My questions are 1) Is the solution I came up with correct? 2) How to find values of $a,b,c$ which guarantee function $f(x,y,z)$ to be strongly convex?
Thanks in advance.