If f(x) = x ⁴ - 2x ² + 3, compute the first and second derivatives:
f'(x) = 4x ³ - 4x
f''(x) = 12x ² - 4
Solve f'(x) = 0 to get the critical points of f(x) :
4x ³ - 4x = 4x (x ² - 1) = 0
→ x = -1, x = 0, x = 1
Check the sign of the second derivative at each of these points:
f'' (-1) = 8 > 0
f'' (0) = -4 < 0
f'' (1) = 8 > 0
These indicate the presence of
• two minima of f (-1) = 2 and f (1) = 2
• one maximum of f (0) = 3
which makes C the correct answer.
