Question

The second order derivative of a sin³α with respect to a cos³α at α = (π/4).

(a) [(4√2)/3a] 

(b) 2

(c) [1/(12a)] 

(d) 0

Answer 1

 The correct option (a) [(4√2)/3a]  

Explanation:

y = a sin³ α ⇒ (dy/dα) = 3a sin² α ∙ cos α.

x = a cos³ α ⇒ (dx/dα) = 3a cos² α (– sin α)

∴ (dy/dx) = – tan α

∴ [(d²y)/(dx²)] = – sec² α ∙ (dα/dx)

∴ [(d²y)/(dx²)] = (– sec² α) ∙ [1/{3a (– sin α) ∙ cos² α}]

∴ [(d²y)/(dx²)] = (1/3a) ∙ [1/{sin α ∙ cos⁴ α}]

∴ [(d²y)/(dx²)]|_α=(π/4) = (1/3a) ∙ [1/{sin (π/4) ∙ cos⁴ (π/4)}]

= (1/3a) ∙ [1/{(1/√2) (1/√2)⁴}] = [(4√2)/3a]