Strict convexity of a function in a given domain

Question

If function $f(\mathbf x)$ is strictly convex at all stationary points in a given domain, $\mathbf x\in[\underline{\mathbf x},\overline{\mathbf x}]$, does it mean that the function may only have local minima in the range and this local minimum is unique in the range i.e., there is only one local minimum in the domain ?

Answer 1

Consider $$ f(x,y)=(x^2-1)(x^2+1)+(y-x^2+1)^2$$ on $[-2,2]\times[-\frac12,\frac12]$.