Question

Find the minimum and maximum values of x+2y where x and y are lengths in the figure attached inside and 0≤x≤10.

**write EXACT answers***

minimum=__________.
maximum=___________.

Answer 1

figure! :D

Answer 2

Is the figure a "closed" figure?

Answer 3

umm not sure.. it doesn't specify :/

do you mean like an interval? :/

Answer 4

Is this calculus?

Answer 5

yes:)

Answer 6

\[
y^2 = 5^2 + (10-x)^2
\]

Answer 7

okay:) using pythagorean theorem right?

Answer 8

\[
f(x) = x + 2y = x + 2\sqrt{5^2+(10-x^2)}
\]

Answer 9

oh so from here, do we need to find the derivative? :/

Answer 10

We need to find critical points. Any place where \(f'(x)\) is \(0\) or undefined.

Answer 11

In this case \(x=0\) and \(x=10\) are critical points as well, simply because they are on the boundary.

Answer 12

ohh okay:)

so here we plug into the original function?
f(0)=
f(10)=
?

Answer 13

or is it a different step? haha not sure if i mixed up the concepts :/

Answer 14

We are only using \(f'(x)\) to find critical points. Once we have all the critical points, we plug them all into \(f(x)\) to find the max and minimum values.

Answer 15

okay
so f(0)=x+2y = 0+2y = 2y?

is it like that?

Answer 16

No, I already told you...

Answer 17

\[
f(x) = x + 2y(x) = x+2\sqrt{5^2+(10-x)^2}
\]

Answer 18

In short: \[
f(x) = x+2\sqrt{5^2+(10-x)^2}
\]

Answer 19

First find \(f'(x)\). Then find \(f'(x) = 0\) and \(f'(x)\) is undefined.

Answer 20

ohh okay oops.. my bad!!

so \[f(0)=0+2\sqrt{25+(10-0)^2 }\]
which equals \[f(0)=0+2\sqrt{125}\] ?

Answer 21

oh derivative first, then plug in?

Answer 22

Yes, derivative first.
Find all critical points first.

Once you have all the critical points, you can plug them into \(f(x)\).

Answer 23

umm so would the derivative be

\[f ' (x) = 0 + \frac{ 2(x-10) }{ \sqrt{x^2-20x+125} }\] ?

Answer 24

is that what i would be plugging in 0 and 10 into?

Answer 25

\[
\frac{d}{dx}x = 1
\]

Answer 26

\[
\frac{d}{dx}2\sqrt{5^2+(10-x)^2} = 2\frac{d}{dx}\sqrt{x^2-20x+125}
\]

Answer 27

Ummm, you can do the rest I think.

Answer 28

ahh not sure what's happening her :/

is it implicit differentiation?

Answer 29

*here

Answer 30

@wio ? :/

Answer 31

sorry, deriving that part is really confusing me :(

Answer 32

No, it isn't implicit because I already gave you an explicit function.

Answer 33

This is the function\[
f(x) = x+2\sqrt{5^2+(10-x)^2}
\]You just have to differentiate this function with respect to \(x\).

Answer 34

ohh okay, thank you!!:)