Explanation

(d)
For $c < 1,\ \ \int_{c}^{1}{\left( 8x^{2} - x^{5} \right)dx = \frac{16}{3}}$
$\Rightarrow \ \frac{8}{3} - \frac{1}{6} - \frac{8c^{3}}{3} + \frac{c^{6}}{6} = \frac{16}{3}$
$\Rightarrow \ c^{3}\left\lbrack - \frac{8}{3} + \frac{c^{3}}{6} \right\rbrack = \frac{16}{3} - \frac{8}{3} + \frac{1}{6} = \frac{17}{6}$
$\Rightarrow \ c = - 1$ satisfy the above equation
For $c \geq 1$, none of the values of $c$ satisfy the required condition that
$\int_{1}^{c}{\left( 8x^{2} - x^{5} \right)dx = \frac{16}{3}}$