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For which $a,b,c\in \mathbb{R}$ is function $$f(x,y)=ax^2+bxy+cy^2$$ convex in $\mathbb{R}^2$? Any ideas on how to approach this problem?

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    Check for which values of $(a,b,c)$ the Hessian matrix of $f$ is positive semi-definite. – 2017-01-18
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    ... so, by Sylvester's criterion, as soon as $a\geq 0$ and $4ac-b^2\geq 0$. – 2017-01-18

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$$\nabla f =\begin{bmatrix} 2ax+by \\ bx+2cy \end{bmatrix} $$

$$\nabla^2f=\begin{bmatrix} 2a & b \\ b & 2c\end{bmatrix}$$

$\nabla^2f$ is positive semidefinite when $2a>0$ and if its determinant is positive.

$a\geq0$ and $(2a)(2c)-b^2=4ac-b^2 \geq 0$.