How do you identify all the critical points for # x^2 – x + y^2 + y = 0#?

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Answer 1 · 2016-11-02T18:48:52

Please see the explanation for a description of how one does it.

Use implicit differentiation, to find the first derivative, #(dy/dx)#:

#2x -1 + 2y(dy/dx) + dy/dx = 0#

Separate #dy/dx# from everything else

#2x -1 + (2y + 1)(dy/dx) = 0#

#(2y + 1)(dy/dx) = 1 - 2x#

#dy/dx = (1 - 2x)/(2y + 1)#

The critical points occur when the first derivative is equal to zero:

#0 = (1 - 2x)/(2y + 1)#

#0 = (1 - 2x)#

#2x = 1#

#x = 1/2#

Find the two values of y where this occurs:

#y^2 + y - 1/2 + 1/4 = 0#

#y^2 + y - 1/4 = 0#

check the discriminant:

#b^2 - 4(a)(c) = 1 - 4(1)(-1/4) = 2#

#y = -1/2 + sqrt(2)/2 and y = -1/2 - sqrt(2)/2#

The critical points are #(1/2,-1/2 + sqrt(2)/2)# and #(1/2,-1/2 - sqrt(2)/2)#