Derieving Derivative #8

So, we want to maximize the value of $x^5y^2$ . We will definitely use the AM-GM inequality. A first thought would have been to directly apply the simple inequality to the numbers ${x, x, x, x, x, y, ,y}$ , but then we will have $5x+2y$ in the LHS, and who's gonna give us that?

So we need some other set of numbers. Now, we're given that $x+y=35$ , so how can we get $x+y$ by a combination of $5$ x's and $2$ y's?

For that, we'll apply the AM-GM inequality to $\frac{x}{5}, \frac{x}{5},\frac{x}{5},\frac{x}{5},\frac{x}{5},\frac{y}{2},\frac{y}{2}$ This will give us $x+y$ , and $x^5y^2$

But wait! Is it the only possible arrangement? We can use it for the set $\frac{x}{10}, \frac{x}{10},\frac{6x}{10},\frac{x}{10},\frac{x}{10},\frac{3y}{4},\frac{y}{4}$ This will also give us $x+y$ , and $x^5y^2$ .

What's the problem in this? The problem is the "equality" case.

In the AM-GM inequality, we attain equality when all the numbers in the set are equal. But in this set, we can't possibly have $\frac{6x}{10}=\frac{x}{10}$ , (x and y are positive)

So, back to the problem, we have the set. Applying the AM-GM inequality, we get:- $\frac{\frac{x}{5}+ \frac{x}{5}+\frac{x}{5}+\frac{x}{5}+\frac{x}{5}+\frac{y}{2}+\frac{y}{2}}{7} \geq (\frac{x^5y^2}{5^52^2})^\frac{1}{7}$ , putting $x+y=35$ , and taking the equality case we get:-

$x^5y^2=4 \times 10^2$ , $2x=5y$ , $x+y=35$ .

Solving them, we get $x=25$ and $y=10$ .

-_- Q.E.D.

Good old calculus can solve it too.

$x+y=35\quad \Rightarrow y = 35 -x$

$\Rightarrow f(x) = x^5y^2 = x^5(35-x)^2 = x^5(1225 - 70x +x^2)$

$\Rightarrow f(x) = x^7 - 70x^6 + 1225x^5$

$f'(x) = 7x^6 - 420x^5 + 6125x^4 = x^4(x-25)(x+35)$

At turning point, $f'(x) = 0$ , then $x = -35, 0, 25$ .

The only acceptable value is $x = 25$ .

It can be found that $f''(25)<0$ .

Therefore, $x^5y^2$ is maximum when $(x,y) = \boxed{(25,10)}$ .

Quick Pythonic solution:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11

biggest = 0
x = 35
y = 0
answers = (0, 0)
while x != 0:
    if x**5*y**2 > biggest:
        biggest = x**5*y**2
        answers = x, y
    x -= 1
    y += 1
print answers