Find two positive numbers with product 200 such that the sum of one number and twice the second number is as small as possible. (calculus approach please)

Question

bahrom7893 · Answer

Find 2 positive numbers with product 200:
$$x * y=200$$The sum of one number and twice the second number is as small as possible
$$f(x)=x+2y$$

bahrom7893 · Answer

Minimize f(x), so find f'(x), see where it's 0 or undefined, and all that.

bahrom7893 · Answer

$$f(x)=x+2y=x+2*\frac{200}{x}$$

bahrom7893 · Answer

$$f(x) = x + \frac{400}{x}=\frac{x*x+400}{x}=\frac{x^2+400}{x}$$

bahrom7893 · Answer

What is the derivative of that function?

anonymous · Answer

Is it $$\frac{ x^2-400 }{ x^2 }$$???

bahrom7893 · Answer

Can you post the steps?

anonymous · Answer

$$f(x)= \frac{ x^2 }{ x }+\frac{ 400 }{ x }

\[=x+400x^{-1}$$

$$f'(x)=1-400x ^{-2}$$

$$=1-\frac{ 400 }{ x^2 }$$

$$=\frac{ x^2-400 }{ x^2 }$$

bahrom7893 · Answer

Correct, now, when is f'(x) equal to 0 or undefined?

anonymous · Answer

x=20?

bahrom7893 · Answer

Actually, when x=+/- 20, and when x = 0. But since you know that the numbers are positive, and it can't be 0, you're right, your answer is 20.

anonymous · Answer

Oh okay, Thanks!

bahrom7893 · Answer

np