Show that x2nbspis a strictly convex function on r without

Question

1. Show that x² is a strictly convex function on R without using any knowledge of the differentiability of the function (that is, using the convex-combination condition of a convex function).

2. Prove that a concave function is quasiconcave. Is the opposite assertion true? That is, if a function is quasiconcave, is it concave? Justify your answer.

3. Show that f(x) = √x is a concave function.

4. Consider the quadratic form f : R²→R defined by

Find the critical points of this function on R² and discuss whether they are maxima, minima or saddle points.