Following is the graph of y = f'(x) and f(0) = 0 .

(a) What type of function y = f'(x) is ? Odd or even?
(b) What type of function y = f(x) is ? Odd or even?
(c) What is the value of `int_(-a)^(a) f(x) dx`?
(d) Has y = f(x) point of inflection?
(e) What is the nature of y = f(x)? Monotonic or non-monotonic?

Explanation

(a) The graph of y = f'(x) is symmetrical about the y-axis, so f(x) is an even function.
(b) f(0) = 0, so f(x) is an odd function [derivative of an odd function is even].
(c) As f(x) is odd, so `underset(-a)overset(a) int f(x) dx = 0`
(d) `f''(x) gt 0 as x lt 0 and f''(x) lt 0 "as " x gt 0`. Therefore, x = 0 is the point of inflexion.
(e) ` f'(x) le 0, forall x, ` so f(x) is always decreasing.