=\cos \mathbf{x}-1+\frac{\mathbf{x}^{2}}{\mathbf{2}}-\frac{\mathbf{x}^{\mathbf{3

The function f(x) = cos x - 1 + x^2/2 - x^3/6 is defined for all real x. Its derivative f'(x) = -sin x + x - x^2/2 has only one critical point at x = 0 (f'(0)=0). The second derivative f''(x) = -cos x + 1 - x gives f''(0)=0, while the third derivative f'''(x) = sin x - 1 gives f'''(0) = -1 ≠ 0. Since the first non-zero derivative at x=0 is odd order, x=0 is a point of inflection, not an extremum. Moreover, f'(x) < 0 for all x ≠ 0, so the function is strictly decreasing, hence no local or global extremum exists.