QJEEM21Ma302a: JEE Main 2021 Mathematics - Class 12 JEEM21Ma302a Solution: Let f : (a, b) R be twice differentiable function such that for …

Question

Verified Solution • 2025–26 Session moderate

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 :

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.

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