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

Correct answer: C. seven roots in (a, b) Let f: (a, b) → R be twice differentiable function such that f(x) = ∫ _ a ^ x g(t)dt for a differentiable function g(x), Now, f(x) = ∫ _ a ^ x g(t)dt Since, f(x) has 5 roots and f ^ ' (x) has atleast 4 roots g(x) has atleast 4 roots Therefore, g ^ ' (x) has atleast 3 roots. Hence, g(x) g ^ ' (x) has atleast 7 roots.

JEE Main 202127 JulyEvening Shift

Exam: JEE Main
Year: 2021
Date: 27 July
Shift: Evening Shift
Subject: Mathematics
Chapter: Application of Derivatives

JEE Main 2021 · 27 July · Evening Shift · Mathematics · Application of Derivatives

Let f: (a, b) → R be twice differentiable function such that f(x) =...

Let

f: \left(a, b\right) \to R

be twice differentiable function such that

f\left(x\right) = {∫}_{a}^{x}g\left(t\right)dt

for a differentiable function

g\left(x\right) .</u> If

f\left(x\right)

= 0has exactly five distinct roots in \left(a, b\right),

then

g\left(x\right)

{g}^{'}\left(x\right)

= 0

has at least : Let f: (a, b) → R be twice differentiable function such that f(x) = ∫ _ a ^ x g(t)dt for a differentiable function g(x)

Options

Atwelve roots in $\left(a, b\right)$
Bfive roots in $\left(a, b\right)$
Cseven roots in $\left(a, b\right)$
Dthree roots in $\left(a, b\right)$

Correct answer

C. seven roots in $\left(a, b\right)$

Step-by-step solution

Let

f: \left(a, b\right) \to R

be twice differentiable function such that

f\left(x\right) = {∫}_{a}^{x}g\left(t\right)dt

for a differentiable function

g\left(x\right),

Let f: (a, b) → R be twice differentiable function such that f(x) = ∫ _ a ^ x g(t)dt for a differentiable function g(x)

Now, $f\left(x\right) = {∫}_{a}^{x}g\left(t\right)dt$

Since, $f\left(x\right)$ has $5$ roots

and ${f}^{'}\left(x\right)$ has atleast $4$ roots

$g\left(x\right)$ has atleast $4$ roots

Therefore, ${g}^{'}\left(x\right)$ has atleast $3$ roots.

Hence, $g\left(x\right){g}^{'}\left(x\right)$ has atleast $7$ roots.

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