Let S be the set of all functions f : [0, 1] → R which are continuous on [0, 1] and differentiable on (0, 1). Then for every f in S, there exists c ∈ (0, 1), depending on f, such that:

The function f(x) = (x + 1)(x - 2)2 has a local minimum at x equal to:

The slope of the normal to the curve y = xx+2 at x = 1 is:

Given that the radius of a circle is increasing at a constant rate of 1 cm/s, what is the instantaneous rate at which the circle's area is increasing when the radius measures 5 cm?