The function f(x) = x^3 , x ∈ R has:

The correct answer is Option (3) → Neither maximum nor minimum value at \(x = 0\)

Here

    \(f'(x) = 3x^2\)
    \(f'(0) = 0\)

So x = 0 is a critical point. But \(f'(x) = 3x^2 \gt 0\) on both sides of 0: the derivative does not change sign, so by the first derivative test x = 0 is neither a point of local maximum nor of local minimum (it is a point of inflexion).

The values agree: for small h > 0,

    \(f(-h) = -h^3 \lt 0 = f(0)\)
    \(f(h) = h^3 \gt 0 = f(0)\)

so f(0) is neither the largest nor the smallest value near 0. Since \(x^3\) takes every real value, f has no maximum or minimum value anywhere on R.

Tip: f′(c) = 0 does not by itself give a maximum or a minimum. Check that f′ changes sign at c; \(x^3\) at 0 is the standard example where it does not.

NCERT (2026-27 edition):
Maxima and Minima