If _0^ f(x) t^2 ,dt = x , , , x , then f ‘ (9) - Vidyadip

MCQ

If $\int\limits_0^{f(x)} {{t^2}\,dt} $ $= x\, \cos\, \pi\, x $, then $f ‘ (9)$

✓
is equal to $-\frac{1}{9}$
B
is equal to $-\frac{1}{3}$
C
is equal to $\frac{1}{3}$
D
is non existent

✓

Answer

Correct option: A.

is equal to $-\frac{1}{9}$

a
$\left. {\frac{{{t^3}}}{3}} \right|_{\,0}^{\,f(x)}$ $= x \cos\, \pi\, x$

$\Rightarrow [f (x)]^3 = 3x\, cos\, \pi x....(1)$

$[f (9)]^3 = - 27\, \Rightarrow f (9)$ $= - 3$

also differentiating $\int\limits_0^{f(x)} {{t^2}\,dt} $ $= x \cos\, \pi \, x$

$[f (x)]^2 · f ‘ (x) = cos\, \pi \, x\, - x\, \pi\, \sin\, \pi \,x$

$\therefore$ $[f (9)]^2 · f ‘ (9) = - 1$

$\Rightarrow f ‘ (9) =$ $- \frac{1}{{{{\left( {f(9)} \right)}^2}}}$ $=-\frac{1}{9}$

$f ‘ (9) =$ $- \frac{1}{9}$

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