Explanation

(d)
Since, $f(x)$ satisfies all the conditions of Rolle’s theorem in [3, 5]
Let $f(x) = (x - 3)(x - 5) = x^{2} - 8x + 15$
Now, $\int_{3}^{5}{f(x)}dx = \int_{3}^{5}\left( x^{2} - 8x + 15 \right)dx$
$= \left\lbrack \frac{x^{3}}{3} - \frac{8x^{2}}{2} + 15x \right\rbrack_{3}^{5}$
$= \left( \frac{125}{3} - 100 + 75 \right) - (9 - 36 + 45)$
$= \frac{50}{3} - 18 = - \frac{4}{3}$