State whether True or False and explain your answer with reasons.

Consider the following function of two variables:

f(x,y) = 2y³ - 6xy - x²

By choosing specific values of x or y, we can construct new functions from f(x,y). For example, f(x,1) is the function constructed from f(x,y) by specifying that y=1. So f(x,1) = 2(1³) - 6x(1) - x² = 2 - 6x - x². Notice that f(x,1) is a function of only one variable, x.

Statement I: The function f(0,y) has a saddle point. Statement II: The function f(9,y) has a local minimum. Statement III: The function f(y,y) has two 2 critical points, of which one is an inflection point.