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Let $f: R^{2} \to R$ be a smooth functions with positive definite Hessian at every point. Let $(a,b) \in R^{2}$ be a critical point of $f$. Then

$a.$ $f$ has a global minimum at $(a,b)$

$b$ $f$ has a local, but not a global minimum at $(a,b)$

$c$ $f$ has a local, but not a global maximum at $(a,b)$

$d$ $f$ has a global maximum at $(a,b)$

I know when, at a stationary point, the Hessian matrix is positive definite then that point is a point of local minima.

In the given problem, the Hessian matrix is positive definite at every point. Then all points should be point of local Maxima.

Any suggestion$?$

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Since $H$ is positive definite, then $f$ is convex. That said, if a convex function admits a critical point $\nabla f(x_0) = 0$, then $x_0$ is a global minimum. So your answer is (a).

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  • $\begingroup$ Is it true that if $H$ is positive definite for every point in $R$ for a function $f$, then that function is convex. $\endgroup$ – Mathsaddict Sep 19 at 11:23
  • $\begingroup$ yes @Mathsaddict, this is necessary and sufficient condition given that $f$ is twice differentiable $\endgroup$ – Ahmad Bazzi Sep 19 at 11:49

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