How select two numbers, x and y, so that their product is as large as possible given that `2x+5y = 70` ?
To answer this question, we need to determine the function we'll be optimizing `F(x,y)`:
`F(x,y) = xy`
Now, to optimize this function, we need to take the derivative and set that function to 0. However, the way we have expressed `F(x,y)` now is useless for optimization! Without adding the other parameters, we simply say "`x` and `y` approach infinity to maximize `F(x,y)` "!
So, let's put the constraint on the problem and get a more useful form of `F(x,y)`:
`2x+5y = 70`
Let's solve for `y` and then substitute that value into `F(x,y)`:
`y = -2/5x+14`
Now, we can have `F(x,y)` become simply a function of x only, `F(x)` simply by substituting the above expression for `y` :
`F(x) = x(-2/5x + 14)=-2/5x^2 + 14x`
Now, we can take the derivative of `F(x)` with respect to `x`:
`(dF(x))/dx = -4/5x+14`
Now, when we optimize `F(x)` we know the derivative will be equal to zero (see link below):
`0 = -4/5x+14`
So, we now solve for x by subtracting 14 and multiplying by -5/4:
`x = 35/2`
Using this value for `x` we can solve for `y`:
`y = -2/5 x + 14 = -7+14 = 7`
So, the two numbers that give the maximum product are `35/2` and `7` giving us a product of:
`F(x,y) = xy = 35/2*7 = 122.5`
And there you have it! I hope this helps.