Suppose that f : [0; 1] -> R is a continous function.
A. Show that the function F: [0; 1] -> R defined by is differentiable in (0; 1), and that for all x E (0; 1) worth F' (x) = f (x);
B. Suppose that f: [0; 1] -> R is additionally differentiable at (0; 1). If for each x E [0; 1] worth the relationship
and f (x) =/= 0 for all x E (0; 1), deduce that f (x) = x for all x E [0; 1].
C. If f> 0, show that
A. Show that the function F: [0; 1] -> R defined by is differentiable in (0; 1), and that for all x E (0; 1) worth F' (x) = f (x);
B. Suppose that f: [0; 1] -> R is additionally differentiable at (0; 1). If for each x E [0; 1] worth the relationship
and f (x) =/= 0 for all x E (0; 1), deduce that f (x) = x for all x E [0; 1].
C. If f> 0, show that