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I have a quadratic function: $f(x) = 24x_1+14x_2+x_1x_2$ and point $x_0 = (2,10)^T$ with $f(x_0) = 208$

And the first question is "give descent direction r in $x_0$" The second question "is f convex in direction r?

How can I do that? I've already determine gradient and Hessean. But which step should be next? According to descent ditrection I fount such formula in the Inet: $r=-\partial{^2}f(x)\partial{f(x)}$ Is that formula correct?

About second question: If Hessean is positive for any x, it means, that f is convex But how to determine whether f convex $\mathbf{in}$ $\mathbf{direction }$?

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(Steepest) descent direction is $-\nabla f(x_0)$. Which is $(-34,-16)^T$ in your case.

To find if $f$ is convex in direction $\vec r$, plug $\vec r$ into the quadratic form associated with the Hessian matrix: $ \vec r^T H \vec r$ gives the second directional derivative in direction $\vec r$. Positive=convex, negative=concave... Here $$(-34,-16)\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} (-34,-16)^T = 2\cdot 24\cdot 16>0$$ Notice that the sign of $\vec r$ does not matter: changing the vector to opposite does not affect convexity in that direction.

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