What are the critical values, if any, of #f(x) = x^3 + x^2 - x #?

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Answer 1 · 2018-03-03T19:36:49

Critical points: #(0.43, -0.49)# and #(-0.77, -2.23)#

We are given: #f(x) = x^3 +x^2-x#

We compute the derivative:
#f'(x) = 3x^2+x-1#

We set the derivative to zero to find critical points:
#f'(x) = 0#
#3x^2+x-1 = 0#

This cannot be factored and solved. We need to use the quadratic equation:

#x = (-b += sqrt(b^2-4ac))/(2a)#
where #a = 3#, #b=1#, and #c=-1#.

This gives:
#x = 0.43# and #x = -0.77#

Substituting these values into #f(x)# gives:
#f(0.43) = -0.49#
#f(-0.77) = -2.23#

Hence the critical points are:
#(0.43, -0.49)# and #(-0.77, -2.23)#