Suppose that f(x, y) is a differentiable function of two variables defined on the entire xy–plane, with the following property, (i) f(x, y) is a strictly increasing function of x for each fixed value...

Suppose that f(x, y) is a differentiable function of two variables defined on the entire xy–plane, with the following property,

(i) f(x, y) is a strictly increasing function of x for each fixed value of y, and (ii) f(x, y) is a strictly increasing function of y for each fixed value of x.

Let g(t) be the function of a single variable defined by g(t) = f(t, t).

A. Give an example of a function f(x, y) that has the indicated property, consider the associated function g(t), and analyze its increasing/decreasing behavior.

B. Make a conjecture regarding the behavior of the function g(t) in the most general setting, and then prove your conjecture.