Convexity and Quasi convexity

Question

i) Is the following function defined with $\mathrm{x}_{1}>0, \quad \mathrm{x}_{2}>0$ quasiconvex? Is it also strictly convex? In each case, provide a full explanation. $$ \mathrm{C}=\mathrm{C}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)=3 \mathrm{x}_{1}^{4}+5 \mathrm{x}_{2}^{2} $$

I guess for this it is quasiconvex ; since incresaing function is always quasiconvex and quasiconcave. Also one can see this by Hessian matrix but what about strictly convex?

Answer 1

If the hessian is positive definite, then it is strictly convex, in this case, the hessian is

$$\begin{bmatrix} 36x_1^2 & 0 \\ 0 & 10 \end{bmatrix}$$ is clearly positive definite over the domain, hence it is strictly convex.