Explanation

(c) Time taken by the body to reach the point $A$ is $t_{1}$ (During upward journey).
The body crosses this point again (during downward journey) after $t_{2},\ ie,\ $the body takes the time $\left( t_{2} - t_{1} \right)$ to come again at point $A.$
So, the time taken by the body to reach at point $B$ (a maximum height).
$t = t_{1}\left( \frac{t_{2} - t_{1}}{2} \right)$
$\lbrack\because Time\ pf\ ascending = Time\ of\ descending\rbrack$
$t = \frac{t_{1} + t_{2}}{2}$
So, maximum height $H = \frac{1}{2}\ gt^{2}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \frac{1}{2}g\left( \frac{t_{1} + t_{2}}{2} \right)^{2}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 2g\left( \frac{t_{1} + t_{2}}{4} \right)^{2}$