Question

Find the critical points of the given function and then determine whether they are local maxima, local minima, or saddle points.

Answer 1

Answer:

$y =-x$ ---- critical point

local minima

Step-by-step explanation:

Given

$f(x,y) = x^2 + y^2 + 2xy$

Required

Determine the critical point

Differentiate w.r.t x

$f_x =2x + 2y$

Differentiate w.r.t y

$f_y =2y + 2x$

Equate both to 0

$2x + 2y =0$

$2y =0-2x$

$2y =-2x$

Divide by 2

$y =-x$ ----- in both equations

Hence:

The critical point is: $y =-x$

Solving (b):

We have:

$f_x =2x + 2y$

$f_y =2y + 2x$

This is represented as:

$D = \left[\begin{array}{cc}2&2\\2&2\end{array}\right]$

Calculate the determinant

$|D| =2 * 2 -2 * 2$

$|D| = 4-4$

$|D| = 0$

The critical point is at local minima