A man on a cycle is travelling on a terrain whose height (h) vs distance (d) can be described by a function $$10d^{4}-5d^{2}$$ in a certain unit (h can be negative). Starting at d = 0, the cycle will have the maximum velocity without pedaling at:​

Correct option is A

Given:
h(d) = 10d⁴ − 5d². Maximum velocity (without pedaling) occurs where potential energy is minimum => find the minimum of h(d) for d ≥ 0.

Formula:
dh/dd = 0 for stationary points.
d²h/dd² to test minima / maxima.

Solution:
dh/dd = d/d d (10d⁴ − 5d²) = 40d³ − 10d = 10d(4d² − 1).
Set dh/dd = 0 => 10d(4d² − 1) = 0 => d = 0 or 4d² = 1 => d² = 1/4 => d = 1/2 (positive root).
Second derivative: d²h/dd² = 120d² − 10.
At d = 1/2: d²h/dd² = 120(1/4) − 10 = 30 − 10 = 20 > 0 => local minimum.
At d = 0: d²h/dd² = −10 < 0 => local maximum. For d ≥ 0 the relevant minimum is at d = 1/2, so the cycle reaches maximum speed there.

Correct Answer: (a) d = 1/2