What Is a Point of Inflection?

A point of inflection is a point where a curve changes concavity: from concave-up to concave-down, or vice versa. In IB Mathematics: Analysis and Approaches (AA), the syllabus uses point of inflexion, but the terms mean the same thing.

How Do You Identify One?

For a twice-differentiable function, concavity is determined by the second derivative:

• If $f''(x)>0$, the graph is concave-up.
• If $f''(x)<0$, the graph is concave-down.

Therefore, $x=a$ gives a point of inflection when (f''(x)) changes sign at $a$. The point on the graph is ((a,f(a))), not merely $a$.

Second-Derivative Test

Worked Examples

For $f(x)=x^3$:

$$f''(x)=6x.$$

At $x=0$, (f''(x)) changes from negative to positive, so the graph changes concavity. Hence, $(0,0)$ is a point of inflection. Since $f'(0)=0$, it is also a .

Now consider $g(x)=x^4$:

$$g''(x)=12x^2.$$

Although $g''(0)=0$, the second derivative is positive on both sides of zero. Therefore, the graph remains concave-up and $(0,0)$ is a minimum, not a point of inflection.

A point of inflection need not be stationary. For $h(x)=x^3+x$, $h''(x)=6x$ changes sign at zero, but . Thus, is a non-stationary point of inflection.

The common misconception is that $f''(a)=0$ automatically proves an inflection point. It only identifies a possible location; you must demonstrate a sign change in (f''), or otherwise establish a change in concavity.

IB Exam Technique

This is AA SL 5.8 content and may appear on Paper 1 or Paper 2. When asked to “show” or “justify,” calculate (f''(x)), identify each candidate value, test the sign on both sides, and state the full coordinate. A table of test values or sign diagram provides clear evidence. On Paper 1, use exact analytic working; on Paper 2, a GDC may support the sign test but does not replace the mathematical conclusion.