STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

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

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

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

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

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

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

$f ‘ (9) =$ $- \frac{1}{9}$