Let f : (a, b) o R be twice differentiable function such that f(x) = \int_a^x {g(t)dt} for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :

Question

Let f : (a, b) 	o R be twice differentiable function such that f(x) = \int_a^x {g(t)dt} for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :

Accepted Answer

Answer: C. seven roots in (a, b) f(x) = \int_a^x {g(t)dt} \Rightarrow f′(x) = g(x) \Rightarrow f′'(x) = g'(x) Given, g(x).g'(x) = 0 \Rightarrow f′(x).f′'(x) = 0 Also given f(x) has exactly 5 roots. So from Rolle's theorem we can say, f′(x) has 4 roots and f′'(x) has 3 roots. 	herefore f′(x).f′'(x) = 0 has 4 + 3 = 7 roots.

Let f : (a, b) $$\to$$ R be twice differentiable function such that $$f(x) = \int_a^x {g(t)dt} $$ for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :

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