Find two positive numbers x and y such that x + y = 60 and xy^3 is maximum.
Find two positive numbers x and y such that x + y = 60 and xy^3 is maximum.

Given:         ……….(i)

Let  P =   [To be maximized]      ……….(ii)

Putting from eq. (i),  in eq. (ii),

P = 

  …..(iii)

Now 

 

 

It is clear that  changes sign from positive to negative as  increases through 45.

Therefore, P is maximum when 

Hence,  is maximum when  and