Optimization

[engineertobe]

The problem is:

Find three positive numbers x, yand z such that the sum of y,z, and 5 times x is equal to 180 and the product of the three numbers is maximum.

I understand the basics of the problem that you have (y+z+5x)=180 but how do you work the problem to maximize the product of the three numbers?

 

[galactus]

Are you familiar with Lagrange multipliers?. You can also use differentiation.

I will use multipliers.

We want to maximize $\displaystyle f(x,y,z)=xyz$ subject to the constraint $\displaystyle g(x,y,z)=5x+y+z=180$

Differentiate:

$\displaystyle \nabla f=yzi+xzj+xyk=\lambda(5i+j+k)$

Equate:

$\displaystyle yz=5\lambda$..........[1]

$\displaystyle xz=\lambda$............[2]

$\displaystyle xy=\lambda$............[3]

Set [1] and [2] equal and we find

$\displaystyle \frac{yz}{5}=xz\rightarrow x=\frac{y}{5}$

From [2] and [3]:

$\displaystyle xz=xy\rightarrow y=z$

sub into the constraint:

$\displaystyle 5(y/5)+y+y=180$

$\displaystyle 3y=180$

$\displaystyle y=60$

Since y=z, then z=60.

and x=60/5=12

The max is $\displaystyle 12\cdot 60\cdot 60$

and the constraint is met $\displaystyle 5(12)+60+60=180$

 

[lookagain]

Is the following not a theorem?


----> The maximum product of n positive real factors occurs when
all of the factors are equal to each other. <----


*If* this is true, then 5x = y = z,


and then 5x = 60.


So x = 12.


And then the maximum = (12)(60)(60).