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[SOLVED] S8.3.7.3. whose sum is a minimum

karush

Well-known member
Jan 31, 2012
2,678
S8.3.7.3.
Find two positive numbers whose product is 100 and whose sum is a minimum
$x(100-x)=100x-x^2=100$

So far

Looks like it's 10+10=20


Doing all my lockdown hw here
since I have no access to WiFi and a PC.
and just a tablet where overkeaf does not work
 
Last edited:

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
S8.3.7.3.
Find two positive numbers whose product is 100 and whose sum is a minimum
$x(100-x)=100x-x^2=100$

So far

Looks like it's 10+10=20


Doing all my lockdown hw here
since I have no access to WiFi and a PC.
and just a tablet where overkeaf does not work
The way you have defined the product implies that $\displaystyle x + y = 100 $, which it almost certainly doesn't.

You are told the product is 100, so $\displaystyle x\,y = 100 \implies y = \frac{100}{x} $.

The sum needs to be minimised, so your sum function is

$\displaystyle \begin{align*} S &= x + y \\ S &= x + \frac{100}{x} \end{align*} $

Now minimise the amount.
 

karush

Well-known member
Jan 31, 2012
2,678
$S'=\left(x + \dfrac{100}{x}\right)' = 1 - \dfrac{100}{x^2}$
So
$S'(0)=10$
 

skeeter

Well-known member
MHB Math Helper
Mar 1, 2012
655
$S'=\left(x + \dfrac{100}{x}\right)' = 1 - \dfrac{100}{x^2}$
So
$S'(0)=10$
No, $S'(0)$ is undefined ...

$S'(x) = 0$ at $x=10$