A) 32
B) \[\frac{32}{3}\]
C) \[\frac{32}{9}\]
D) None of these
Correct Answer: C
Solution :
Given, \[f(x)=\frac{1}{{{x}^{2}}}\int_{4}^{x}{(4{{t}^{2}}-2f'(t)\}dt}\] On differentiating both sides, we get \[f'(x)=\frac{1}{{{x}^{2}}}[4{{x}^{2}}-2f'(x)]\] \[-\frac{2}{{{x}^{3}}}\int_{4}^{x}{[4{{t}^{2}}-2f'(t)]dt}\] \[\Rightarrow \]\[f'(4)=\frac{1}{16}[64-2f'(4)]-0\] \[\Rightarrow \]\[f'(4)\left( 1+\frac{1}{8} \right)=4\]\[\Rightarrow \]\[f'(4)=\frac{32}{9}\]You need to login to perform this action.
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