Michael J. answered 04/05/15
Best Afterschool Tutor to Prepare You For Your Regents
To find the critical points, we must set the derivative of f(x) equal to zero. The critical points are the location of maximum and minimum points of a graph. The max and min have lines whose slope is zero hat are tangential to them.
Let's make f(x) easier to find the derivative of.
f(x) = 10(x - 1)(x - 1)e-x
f(x) = 10e-x(x2 - 2x + 1)
d/dx[f(x)] = 0
-10e-x (x2 - 2x + 1) + 10e-x (2x - 2) = 0
-10e-x [(x2 - 2x + 1) - (2x - 2)] = 0
-10e-x (x2 - 2x + 1 - 2x + 2) = 0
-10e-x (x2 - 4x + 3) = 0
Set the term in parenthesis to zero.
x2 - 4x + 3 = 0
(x - 3)(x - 1) = 0
x = 3 and x = 1
Our critical points are x = 1 and x = 3.