The function f(x) = 2x^4 - 4x^2 + 5 , x ∈ R , has

The correct answer is Option (3) → a local maximum at \(x = 0\) and local minima at \(x = \pm 1\)

The critical points are where \(f'(x) = 0\):

    \( f'(x) = 8x^{3} - 8x \)

    \( = 8x(x - 1)(x + 1) \)

So the critical points are x = 0, x = 1 and x = −1.

By the second derivative test, with

    \( f''(x) = 24x^{2} - 8 \)

    \( f''(0) = -8 \lt 0 \), so x = 0 is a point of local maximum
    \( f''(\pm 1) = 24 - 8 = 16 \gt 0 \), so x = ±1 are points of local minimum