Question

If $\underset{0}{\overset{f(x)}{\int }}{t}^{2}dt=x\cos \pi x$, then $f^{\prime }(9)$

• A. 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 1

Answer: (A)

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)
$f^{\prime }(9)=-1/9$