Question

Please help! It's calculus and I know it involves finding critical points by using the derivative but I'm so confused! Thank you so much to anyone who can help me out!


the sum of two nonnegative numbers is 20. Find the numbers if
(a) the sum of their squares is as large as possible; as small as possible

(b) one number plus the square root of the other is as large as possible; as small as possible

Answer 1

let x , y be the two numbers

we want to maximize x^2 + y^2 , and minimize x^2 + y^2 , given that the sum x + y = 20

Answer 2

first thing i would do is get the expression x^2 + y^2 in one variable

Answer 3

Hey you helped me with physics a couple of weeks ago! Nice to hear from you again! How would I get that expression in one variable?

Answer 4

thanks.
we are given that x + y = 20, so we can solve for y

Answer 5

and then substitute

Answer 6

ah okay that makes sense. So y=20-x
so
\[x^2 + (20-x)^2\]
Where do I need to go from there?

Answer 7

so now we have an expression or function in one variable
f(x) = x^2 + (20-x)^2

to maximize it or minimize it, we solve f ' (x) = 0 , also you can test the endpoints of the domain

Answer 8

since the two numbers are nonnegative
the domain for x is [0 , 20]

Answer 9

Oh okay so that equation is my new f(x) so I find the derivative and find critical points. Okay that should solve A, then B is the same concept. I just need to substitute in y to make it all one variable. Okay that makes a lot of sense. Thanks so much again! I've got a whole page of these to do, but they should be easy since I got the concept!

Answer 10

:)