Question

divide the number 84 into two parts such that the product of one part and the square of the other is maximum

Answer 1

let x and (84-x) are two part of 84
a/c to question,
x^2.(84-x)=f(x)

f(x)=84x^2-x^3
diffrentiate w.r.t x
f'(x)=168x-3x^2
f'(x)=0
x(168-3x)=0
x=0 and x=168/3=56
again differentiate w.r.t x
f"(x)=168-6x
put x=56 in f"(x) you find f"(x)<0
it means at x=56 product is maximum.
so,
56 and 28 are two numbers