Find 3 numbers to maximize product given specific restriction 2x+y+z=210 (all pos)?

Question

Find three positive numbers x, y and z such that the sum of y, z, and 2 times x is equal to 210 and the product of the three numbers is maximum.

How would I go about solving this?

I tried to do sub so that z=210-y-2x
then x * y * (210-y-2x) is maximized

so I took gradient

(-(4x+y-210)y,-2(y+x-105)x) and tried to solve each fx,fy=0 but i got a very odd answer

In response to first answer

35 turned out to be the correct value for x but y and z were both 70 not 69 and 71

Answer 1

Your problem did not state that the three numbers are different. You can approach this problem the way you started, finding the function f(x,y). Then for a local maximum of the surface, the conditions come down to ∂f(x,y)/∂x = 0 and ∂f(x,y)/∂y = 0. You take the partial derivatives and will get two equations in x and y, which can be solved. The result is x=35, y=70, z=70.

Answer 2

2x + y + z =210

F(x, y, z) = xyz = xy(210 - 2x - y)
= 210xy - 2x²y - xy²
F'(x, y, z) = 210y + 210xdy/dx - 2x²dy/dx - 2xydy/dx
= 0 for stat points

ie 2x²dy/dx - 210xdy/dx + 2xydy/dx = 210y - 4xy - y²

2x(x + y - 105)dy/dx = y(210 - 4x - y)
= 0 if y = 0 or 4x + y = 210
clearly when y = 0 there is a minimum in the product as xyz = 0

if 4x + y = 210 then z = 2x as 2x + y + z = 210

Now if 4x + y = 210
y = 210 - 4x
and f(x, y, z) = 2x²y = 2x²(210 - 4x) = 420x² - 8x³
f'(x. y, z) = 840x - 24x²
= 0 for stat pys
24x(35 - x) = 0

when x = 0 there is a minimum
when x = 35 there is a maximum

So x = 35, z = 70 (= 2x) and y = 70 (= 210 - 4x)

and xyz max = 35 * 70 * 70 =171500

Just looking at above solution. You did not say that the numbers were either integers or had to be different.

If so the next closest thing to 35 * 70 * 70 is 35 * 69 * 71 (where 2x = 70)

Answer 3

for product of 3 numbers to be maximized, they hv to be consequtive. (u can play around with numbers if u don't believe)
for 210, the only 3 consequtive numbers tt can suit is 69 + 70 + 71.
since one of the terms must be 2x, so 2x will be the only even no. here. hence x = 35. while y and z are 69 and 71.
this should be max.

Answer 4

Let X = 2x, Y = y and Z = z, then problem is to maximize

(X/2) Y Z, which is the same as maximizing XYZ, subject to

X + Y + Z = 210

This occurs when X = Y = Z (by symmetry, or by calculus), that is when

2x = 70, y = 70 and z = 70, or

x = 35, y = 70 and z = 70.

Answer 5

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