Local Approximation of Real Valued Functions

Question

I'm unsure where to begin. Any guidance would be greatly appretiated. Suppose that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ has continuous second order partial derivatives, and at the origin (0,0) suppose that $\frac{\partial f}{\partial x}(0,0)=0$ and $\frac{\partial f}{\partial y}(0,0)=0$. Let $h$ be a nonzero point in the plane $\mathbb{R}^2$ and suppose that $\langle\bigtriangledown^2f(0,0)h,h\rangle>0.$ Use single-variable theory to prove that there is some positive number $r$ such that $f(th)>f(0,0)$ if $0<|t|<r.$

Answer 1

Hint: Calculate the second derivative of $g(t)=f(th)$, and recall the criteria for local extremal points in one variable.