Eeek! help please:)

[pinkie]

The product of 2 positive numbers is 16. Find the 2 numbers such that
a) the sum is a minimum
b) the sum of one and the square of the other is minimum

i think i got the answer to the first part but im not sure lol. Hopefully someone can help me out! Thanks ♥

 

[soroban]

Hello, pinkie!

The product of two positive numbers is 16.
Find the two numbers such that
a) the sum is a minimum
b) the sum of one and the square of the other is minimum

Let \(\displaystyle x\) be one number and \(\displaystyle y\) be the other.

Then we have: \(\displaystyle \,xy\,=\,16\;\;\Rightarrow\;\;y\,=\,\frac{16}{x}\)

(a) The sum is a minimum.
\(\displaystyle \;\;\:\)We have the function: \(\displaystyle \:S\;=\;x\,+\,y\;=\;x\,+\,\frac{16}{x}\;=\;x\,+\,16x^{-1}\)
\(\displaystyle \;\;\;\) Minimize it!


(b) The sum of and the square of the other is a minimum.
\(\displaystyle \;\;\;\)We have the function: \(\displaystyle \:S\;=\;x^2\,+\,y\;=\;x^2\,+\,\frac{16}{x}\;=\;x^2\,+\,16x^{-1}\)
\(\displaystyle \;\;\;\) Minimize it!