Find all relative extrema of the function f(x)= x^4 - 4x^2 + 2 by using the Second Derivative Test where applicable.

Question

Find all relative extrema of the function f(x) = x⁴ - 4x²+ 2 by using the Second Derivative Test where applicable.

Answer 1

f(x) = x⁴ - 4x² + 2

f'(x) = 4x³ - 8x = 4x(x² - 2) = 0 when x = 0, √2, -√2

f"(x) = 12x² - 8

f"(0) = -8 < 0 So, f has a rel max at (0, f(0)) = (0, 2)

f"(√2) = f"(-√2) = 16 > 0 So, f has rel minima at (√2, -2) and at (-√2, -2)