list the critical points of f x x4 2x2 1 use second deriva

list the critical points of : f (x) = x^4 -2x^2 +1.

use second derivative test to determine if each critical point is a minimum or a maximum

Solution

f\'(x) = 4x^3 - 4x f\' = 0 => 4x(x^2-1) = 0 => x =0, x = -1, x = +1 are the critical points f\" = 12x^2 - 4 f\" at x = 0 = -4 , so point of maxima at x = 1, = positive, so minima at x = -1, positive, so minima