"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 :"

Question

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."

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 :

Solution

About this question

Drill 25 more like these. Every day. Free.

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 :

Solution

About this question

More Definite Integration questions

Drill 25 more like these. Every day. Free.