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Q.
If $\int\limits_0^{f(x)} t^2 d t=x \cos \pi x$, then $f^{\prime}(9)$
Integrals
Solution:
On differentiating both sides
$[f( x )]^2 f^{\prime}( x )=\cos \pi x -\pi x \sin \pi x$
$[f(9)]^2 f^{\prime}(9)=-1$.....(i)
Also $\left[\frac{ t ^3}{3}\right]_0^{f(x)}=x \cos \pi x \Rightarrow \frac{[f(x)]^3}{3}=x \cos \pi x$
$[f(9)]^3=-27 \Rightarrow f(9)=-3$.......(ii)
from (i) & (ii)
$f^{\prime}(9)=-1 / 9$