Question

What is the smallest value of \[x^2 + 4xy + 5y^2 - 4x - 6y + 7\]

Answer 1

Use the same method. Complete the squares.

Answer 2

But this requires a little more thoughtfulness as to what squares can (or should) be completed.

Answer 3

it does :( thinking time :-D

Answer 4

I'm sure there's an easy way out of this one if I think hard enough. This is from Spivak

Answer 5

and it's still the first chapter

Answer 6

if I can do this one I will be a master at completing squares :-D.... but this might take half an hour :(

Answer 7

Spivak is really expecting the reader to make a huge leap of intuition here :(

Answer 8

i don't like that xy term in there :(
we could have just used our algebra skills

Answer 9

you know write it in that form where you find the center of the ellipse and then look at the up and down radius to figure out the lowest point

Answer 10

but this is just in the beginning of the book, where I'm not supposed to know what an ellipse is yet.

Answer 11

oh
beginning of what book?

Answer 12

Calculus 3rd, Michael Spivak

Answer 13

i thought they talked about ellipse in algebra
so this book assumes you don't know the standard form for an ellipse

Answer 14

I don't really know what it's assuming out of me :-P

Answer 15

lol there is no way to tell i don't think

Answer 16

but this is a chapter about the basic laws of addition and multiplication, so we have to apply those somewhere here.

Answer 17

You can use whatever you like about geometry. This book is not assuming you've never seen mathematics before.

Answer 18

lol

Answer 19

I think the only geometry in the chapter was the part about absolute values....

so how do we do the ellipse trick here?

Answer 20

we can't use differentiation here since that's chapter 6 or something

Answer 21

like i said i can't remember with that xy term
i will be right back
i have to get the clothes out of the dryer

Answer 22

brb

Answer 23

\[x^2 + 4xy + 5y^2 - 4x - 6y + 7\]\[=x^2 + 4x(y-1) + 5y^2 - 6y + 7\]\[=x^2 + 4x(y-1) + 4(y-1)^2 + 5y^2 - 6y + 7 - 4y^2 +8y - 4\]\[=(x+2(y-1))^2 + y^2 + 2y + 3\]\[=(x+2(y-1))^2+y^2 + 2y + 1 + 3 -1\]\[=(x+2(y-1))^2 + (y+1)^2 +2\]

when x is -4 and y is -1 it seems the expression is equal to 2

Answer 24

when x = 4, y = -1

Answer 25

oops when x is 4

Answer 26

you've got it.

Answer 27

now do the problem before this one by yourself

Answer 28

I did it a little earlier.

Answer 29

good

Answer 30

I think the answer was -9/4 something