For the first problem, The definite integral from 0 to c is [f(c) - f(0)] / (c - 0) since the function is continuous in the interval 0 to c.
Since f(a) = f(b) = 1, then by the Mean Value Theorem, there is a point somewhere in the middle two values of x that f'(x) = 0 since the curve has a degree of more than 2.
