Critical Point(s) of a Multivariable Function

1. Mar 25, 2013

johnhuntsman

1. The problem statement, all variables and given/known data
Find the critical points of f.
$$f(x,y)=2+\sqrt{3(x-1)^2+4(y+1)^2}$$

2. Relevant equations
For fx(x,y) I get:
$$f_x(x,y)=0+\frac{1*6(x-1)}{2\sqrt{3(x-1)^2+4(y+1)^2}}=\frac{3(x-1)}{\sqrt{3(x-1)^2+4(y+1)^2}}$$
For fy(x,y) I get:
$$f_y(x,y)=0+\frac{1*8(y+1)}{2\sqrt{3(x-1)^2+4(y+1)^2}}=\frac{4(y+1)}{\sqrt{3(x-1)^2+4(y+1)^2}}$$

3. The attempt at a solution
Solving both of these for x and y when set equal to 0, gets me x = 1 and y = -1. However neither of these functions exist when x and y equal those values. Does the original function still have a critical point at (x,y) = (1,-1)?

Additionally, when I put the function into Wolfram Alpha it says that it has no critical points.

Last edited: Mar 25, 2013
2. Mar 26, 2013

voko

What is the definition of the critical point?