Show that semi-vertical angle of a cone of maximum volume and given slant height is cos⁻¹(1/√3)

Let radius, height and slant height of cone be r, h and l respectively.
Therefore, r² + h² = l²
Volume of cone (V) = (1/3)πr²h
∴V = (π/3)h(l² - h²)
⇒(π/3)[l²h - h³]
Differentiating w.r.t to h, we get
dV/dh = (π/3)(l² - 3h²)
When dV/dh = 0
⇒h = l/√3
d²V/dh² = -2πh
d²V/dh² = -2π(l/√3)
⇒ -2πl/√3 < 0
Therefore at h = l/√3, volume is maximum.
cos α = h/l = 1/√3
∴ α = cos⁻¹(1/√3).