JEE MAIN - Mathematics (2021 - 27th July Evening Shift - No. 15)
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 :
twelve roots in (a, b)
five roots in (a, b)
seven roots in (a, b)
three roots in (a, b)
Explanation
$$f(x) = \int_a^x {g(t)dt} $$
$$ \Rightarrow $$ 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.
$$ \Rightarrow $$ 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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