Let F(x) = ∫ _ x ^ x ^ 2 + (π)/(6) 2 ^ 2 tdt for all x ∈ ℝand f:...

${F}^{′}$\left(a\right) + 2 = {∫}_{0}^{a}f\left(x\right)dx$<br />$⇒ {F}^{′′}\left(a\right) = f\left(a\right)$...(i)<br />$∵ F\left(x\right) = \mathop{∫}_{x}^{{x}^{2} + \dfrac{\pi}{6}}2\cos^{2} tdt$<br />$∴ {F}^{′}\left(x\right) = 2\cos^{2} \left({x}^{2} + \dfrac{\pi}{6}\right) . 2x - 2\cos^{2} x$<br />${F}^{′′}\left(x\right) = 4\cos^{2} \left({x}^{2} + \dfrac{\pi}{6}\right) - 8xcos \left({x}^{2} + \dfrac{\pi}{6}\right) . \sin \left({x}^{2} + \dfrac{\pi}{6}\right)2x + 4cos xsin x$<br />$f\left(0\right) = {F}^{′′}\left(0\right) = 4 \times \dfrac{3}{4} = 3