Following is the graph of y = f' (x) , given that f(c) = 0. Analyse the graph and answer the following questions.
(a) How many times the graph of y = f(x) will intersect the x - axis?
(b) Discuss the type of roots of the equation f (x) = 0, ` a le x le b`.
(c) How many points of inflection the graph of ` y = f(x), a le x le b`, has?
(d) Find the points of local maxima/minima of` y = f(x), a lt x b`.
(e) How many roots equation ` f''(x) = 0` has?

Explanation

(a) ` f'(x) le 0, forall x in [a, b], ` so f(x) is a decreasing function and `f(c) = 0 rArr f(x)` is a decreasing function and ` x (c) = 0 rArr f(x) ` cuts x - axis once when x = c.
(b) We note that f(c) = 0, f'(c) = 0. Also tangent to f'(x) at x = c is y = 0. So f''(c) = 0 Therefore, x = c is repeated root of third order. So the equation f(x) = 0 has at least three repeated roots.
(c) We have f''(c) = 0 . So the graph of y = f (x) has one point of inflection at x = c.
(d) As f (x) is a decreasing function for all ` x in (a, b) , f(x)` has no local maxima or minima.
(e) ` f''(c) = 0 rArr c ` is a root of f''(x) = 0.