Question

classify all critical points
g(x, y) = 1 − x^(2) − x − 2y^(2) + y

Answer 1

Find where the gradient of g

Answer 2

The gradient is
{-1 - 2 x, 1 - 4 y}
set it up ={0,0} and solve for x and y

Answer 3

evaluating the first and second order partial derivatives would be a good start

Answer 4

You find
\[
\left\{\left\{x\to -\frac{1}{2},y\to \frac{1}{4}\right\}\right\}
\]

Answer 5

The hessian determinant is always 8. Now study the sign of the second derivative with respect to x

Answer 6

The second derivative is always equal to -2, so (-1/2, 1/4) is a local max

Answer 7

Here is a plot

Answer 8

Are you still with me?

Answer 9

yeah wait why when its X alone does it just stay as 1?

Answer 10

meant to delete why

Answer 11

I did not understand your question

Answer 12

where did you get -1 from in -1-2x

Answer 13

The partial derivative of g with respect to x

Answer 14

so X equals 1? it doesn't just get cancelled out?

Answer 15

what is the derivative of − x^(2) − x with respect to x
It is
-2 x -1

Answer 16

okay that makes sense, i was a little confused if x = -1

Answer 17

You might review your differentiation skills. You can do that on my site
http://www.saab.org/calculus.cgi

Answer 18

when solving for the partial derivative

Answer 19

Do you understand it now?