Mathematics

The key is finding a function whose derivative touches zero at a point but doesn't change sign there—so the function keeps increasing (or decreasing) right through it. Think of $f(x) = x^3$: the tangent line is perfectly horizontal at $x=0$, yet the function is strictly increasing everywhere. The derivative $f'(x) = 3x^2$ equals zero at $x=0$, but for $x < 0$ the function is below zero and for $x > 0$ it's above, so there's no local max or min, just an inflection point where the curve changes concavity.

The key is finding a function whose derivative touches zero at a point but doesn't change sign there—so the function keeps increasing (or decreasing) right through it. Think of $f(x) = x^3$: the tangent line is perfectly horizontal at $x=0$, yet the function is strictly increasing everywhere. The derivative $f'(x) = 3x^2$ equals zero at $x=0$, but for $x < 0$ the function is below zero and for $x > 0$ it's above, so there's no local max or min, just an inflection point where the curve changes concavity.