1.

Following is the graph ofy = f' (x) , given thatf(c) = 0. Analyse the graph and answer the following questions. (a) How many times the graph ofy = f(x) will intersect thex - 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 ofy = 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 equationf''(x) = 0 has?

Answer»

Solution :(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 rArrf(x) ` cutsx - axis once when x = c.
(b) We NOTE thatf(c) = 0, f'(c) = 0. Also tangent tof'(x)atx = cis y = 0. So f''(c) = 0Therefore, x = c is repeated root of third order. So theequation f(x) = 0 has at least three repeated roots.
(c) We have f''(c) = 0 . So thegraph of y = f (x)has ONE point of inflection at x = c.
(d) Asf (x) is a decreasing function for all` x in (a, b) ,f(x)` has no local maxima or minima.
(E) ` f''(c) = 0 rArrc `ISA root off''(x) = 0.


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