Maximum Value?

Question

Let "u" and "v" be two non-negative numbers, whose sum is 10. Find the maximum value of "u^2*v". Determine "u" and "v" when the maximum is attained.

Answer 1

We have v = 10 - u. If p = u^2 v, then, p = u^2(10 - u) = 10 u^2 - u^3. Hence,

dp/du = 20 u - 3u^2 = u(20 - 3u). Therefore,

dp/du < 0 if u is in (-oo, 0) U (20/3 , oo)
dp/du = 0 if u = 0 or u = 20/3
dp/du >0 if u is in (0, 20/3)

So, p decreases on (-oo 0), attains a minimum at u = 0, increases on (0, 20/3), attains a maximum at u = 20/3 and decreases on (20/3, oo)

But we must have u in (0, 10) and this interval contains 20/3. So, p has a global maximum at u = 20/3.

The values are u = 20/3 and v = 10 - 20/3 = 10/3

Answer 2

u + v = 10
So:
v = 10-u (duh...)
so,
u^2(10-u)
u^20-2u
and the maximum of that formula should be easy to find on your ...fancy calculator(dont know what the english word is for a calculator that can plot charts), persuming you have one...
good luck

Answer 3

Well, if they can only be integers, than it's 147 (7^2*3)

Answer 4

Max value of (u^2)v = 148.148(3 d.p)
For Max Value:
U = 6.667
V = 3.333
(3 d.p)