Let f:(a,b)R be a twice differentiable function such that f(x)=_a^xg(t),dt for a different…

Question

Let $f:(a,b)	o\mathbb{R}$ be a twice differentiable function such that $f(x)=\int_a^xg(t)\,dt$ 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

$$f(x) = \int_a^x {g(t)dt} $$

$$ \Rightarrow $$ 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.

$$ \therefore $$ f'(x).f''(x) = 0 has 4 + 3 = 7 roots.

Solutions · 1