If F(x) = 1 x^2 _4^x (4 t^2 - 2F'(t)) ,dt, then F'(4) equals - Vidyadip

MCQ

If $F(x) = \frac{1}{{{x^2}}}\int_4^x {(4{t^2} - 2F'(t))\,dt,} $ then $F'(4)$ equals

A
$32$
B
$\frac{{32}}{3}$
✓
$\frac{{32}}{9}$
D
None of these

✓

Answer

Correct option: C.

$\frac{{32}}{9}$

c
(c) We have $F(x) = \frac{1}{{{x^2}}}\int_4^x {(4{t^2} - 2F'(t))dt} $

$\therefore \,\,F'(x) = \frac{1}{{{x^2}}}\left( {4{x^2} - 2F'(x)} \right) - \frac{2}{{{x^3}}}\int_4^x {(4{t^2} - 2F'(t))dt} $

==> $F'(4) = \frac{1}{{16}}[64 - 2F'(4)] - 0 $

$\Rightarrow F'(4) = \frac{{32}}{9}$.

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