Finding Turning Points: Maxima and Minima

1. What are Stationary Points?

A stationary point is a point on a curve where the gradient is zero. At these points, the tangent to the curve is a horizontal line. These are also called turning points because they are where the curve changes direction from increasing to decreasing, or vice versa.

There are two main types you need to know for your exam:

• Maximum point: The "peak" of a section of the curve.
• Minimum point: The "trough" of a section of the curve.

Let's watch a short video that introduces this concept visually.

2. The Method: Finding and Classifying Stationary Points

To solve these problems efficiently in an exam, you can follow a reliable, five-step process. This process first locates the points and then determines their nature.

The 5-Step Strategy

1. Find the Gradient Function: Differentiate the curve's equation $y=f(x)$ to find $\frac{dy}{dx}$.
2. Find Stationary x-coordinates: Set the gradient function to zero ($\frac{dy}{dx}=0$) and solve the resulting equation for $x$. This often involves solving a linear or quadratic equation.
3. Find Full Coordinates: Substitute the $x$-values you found back into the original equation $y=f(x)$ to find the corresponding $y$-coordinates.
4. Find the Second Derivative: Differentiate the gradient function ($\frac{dy}{dx}$) to get the second derivative, $\frac{d^{2}y}{d{x}^2}$.
5. Classify the Points: Substitute each stationary $x$-coordinate into the second derivative, $\frac{d^{2}y}{d{x}^2}$, and check the sign of the result.
   • If $\frac{d^{2}y}{d{x}^2}<0$ (negative), the point is a Maximum.
   • If $\frac{d^{2}y}{d{x}^2}>0$ (positive), the point is a Minimum.

Let's walk through this process.

Steps 1-3: Locating the Coordinates

The first part of the process is finding where the stationary points are.

Steps 4-5: Classifying the Points with the Second Derivative

Now that we have the coordinates, we need to determine if they are maximums or minimums. For exam purposes, the most efficient way is the Second Derivative Test.

The second derivative, $\frac{d^{2}y}{d{x}^2}$, tells us about the rate of change of the gradient.

• At a maximum point, the gradient is decreasing (it goes from positive, to zero, to negative). A decreasing gradient means $\frac{d^{2}y}{d{x}^2}$ is negative.
• At a minimum point, the gradient is increasing (it goes from negative, to zero, to positive). An increasing gradient means $\frac{d^{2}y}{d{x}^2}$ is positive.

Let's watch a video that explains this test and applies it.

3. Consolidation and Practice

Let's solidify this method with a concise written summary and a full worked example. The "second derivative test" is the method you should prioritize for exams as it is generally quicker and less prone to the small arithmetic errors you're working on eliminating.

Now, try a full question yourself, applying the 5-step strategy.

Conclusion

Excellent work. You have now learned one of the most powerful applications of differentiation. Being able to find and classify turning points is a skill that is tested frequently and carries significant marks.

Key Takeaways:

• Stationary (or turning) points occur where the gradient of the curve is zero, i.e., $\frac{dy}{dx}=0$.
• To find the coordinates, solve $\frac{dy}{dx}=0$ for $x$, then substitute these $x$-values back into the original equation to find $y$.
• To determine the nature of a stationary point, use the second derivative test:
  • Find $\frac{d^{2}y}{d{x}^2}$.
  • Substitute the $x$-coordinate of the stationary point.
  • If the result is positive, it's a minimum.
  • If the result is negative, it's a maximum.

Preview of the Next Lesson:
We have spent the last few lessons on differentiation—the process of finding a rate of change. Now, we will explore the reverse process: integration. In the next lesson, we will learn how to integrate polynomial functions. This will open the door to another major application of calculus: finding the area under a curve.