QJEEM19Mac7f7: JEE Main 2019 Mathematics - Class 12 JEEM19Mac7f7 Solution: If f : R → R is a differentiable function and f(2) = 6, then…

Question

Verified Solution • 2025–26 Session moderate

If f : R

\to

R is a differentiable function and f(2) = 6,
then

lim_{x \to 2} \frac{\int_{6}^{f (x)} 2 t d t}{(x - 2)}

is :-

Accepted Answer

lim_{x \to 2} \frac{\int_{6}^{f (x)} 2 t d t}{(x - 2)}

This is

\frac{0}{0}

form so we use L – Hospital Rule.

=

lim_{x \to 2} \frac{2 f (x) . f^{'} (x) - 0}{1}

=

2 f (2) . f^{'} (2)

= 12f'(2)

Note : Newton - Leibnitz rule of differentiation under integration

\frac{d}{d x} \int_{α (x)}^{β (x)} f (u) d u

=

f (β (x)) β^{'} (x) - f (α (x)) α^{'} (x)

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