If , then is

Engineering

Mathematics

Leibnitz Rule Derivative of Anti Derivatives

Question

If $\int_{0}^{x} f (t) dt = x^{2} + \int_{x}^{1} t^{2} f (t) dt$ , then $f' (\frac{1}{2})$ is

$\frac{24}{25}$

$\frac{4}{5}$

$\frac{6}{25}$

$\frac{18}{25}$

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Differentiability we get f (x) = 2x – x² f (x)

$f (x) = \frac{2 x}{1 + x^{2}} \Rightarrow f " (x) = 2 \frac{(1 - x^{2})}{{(1 + x^{2})}^{2}}$

$f' (\frac{1}{2}) = \frac{24}{25}$