How Do You Find Where the Gradient of a Curve Equals Zero?

Differentiate the function and solve $f'(x)=0$. The resulting $x$-values are where the curve has zero gradient, called stationary points.

A zero derivative means the tangent is horizontal at that input. However, the curve may turn, or it may continue increasing or decreasing through a stationary point of inflexion.

This is part of SL 5.6, Stationary points, local maximum and minimum points, studied by both SL and HL Mathematics: Applications and Interpretation students.

Method

For a curve $y=f(x)$:

1. Find the derivative (f'(x)), which represents the gradient of the curve.
2. Set the gradient equal to zero: $f'(x)=0.$
3. Solve this equation, algebraically or using a graphic display calculator (GDC).
4. Substitute each solution into $f(x)$ to obtain the coordinates of the stationary points.
5. Check how the sign of (f'(x)) changes around each point.

Worked Example

Let $f(x)=x^3-3x^2+2.$

Differentiate: $f'(x)=3x^2-6x=3x(x-2).$

Set the derivative equal to zero: $3x(x-2)=0.$

Therefore, $x=0$ or $x=2$. Substitution gives $f(0)=2$ and $f(2)=-2$, so the stationary points are and .

Here, (f'(x)) changes from positive to negative at $x=0$, so $(0,2)$ is a local maximum. It changes from negative to positive at $x=2$, so $(2,-2)$ is a local minimum.

For this example, testing one value in each interval $x<0$, $0<x<2$, and $x>2$ gives derivative signs $+$, (-), and , respectively. This confirms both classifications.

On a restricted domain, also check the endpoints when finding the greatest or least value. A local stationary point is not necessarily the global maximum or minimum.

Exam Technique

A common misconception is that every solution of $f'(x)=0$ is a maximum or minimum. It is only guaranteed to be a stationary point; classification requires a sign check or inspection of the graph. In AI Paper 1 or Paper 2, show the equation $f'(x)=0$ before giving GDC solutions, retain full calculator precision, and state coordinates rather than only the -values when points are requested.