Explanation

(d)
Given $f(x)$satisfy Rolle’s theorem in [3, 5].
$\therefore f(x)$is continuous in [3,5]
And $f(x)$is differentiable in]3,5[
And $f(a) = f(b) = 0ie,f(3) = f(5) = 0$
Let $f(x) = (x - 3)(x - 5)$
$\therefore f(x) = x^{2} - 8x + 15$
$\therefore\int_{3}^{5}{\left( x^{2} - 8x = 15 \right)dx = \left\lbrack \frac{x^{3}}{3} - \frac{8x^{2}}{2} + 15x \right\rbrack_{3}^{5}}$
$= \left( \frac{50}{3} - 18 \right) = - \frac{4}{3}$