Prove that is an increasing function of θ on

OR

Show that semi-vertical angle of a cone of maximum volume and given slant height is

⇒

⇒

⇒

⇒

Notice that the denominator is strictly greater than zero.
On the interval [0, π/2] we have that 0 ≤ cos(θ ) ≤ 1. Therefore, the numerator is always strictly greater than 0.
Therefore, we have that y' > 0 for all θ ∈ [0, π/2]. Therefore, y is strictly increasing on [0, π/2].

OR

Let slant height of the cone to be l,

r = lsinθ

h = lcosθ

Now,

For maximum volume,

Therefore,

2sinθcos²θ = sin³θ

2cos²θ = (1 – cos²θ)

3cos²θ = 1

Hence, Proved.