# Problem on Hessian matrix.

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$$?$$

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