Differentiation is essential for understanding how functions behave. In this section, the focus is on stationary points, which are crucial in analysing the nature of functions.

**Stationary Points, Increasing and Decreasing Functions**

**Increasing Functions:**When f'(x) > 0, the function is increasing.**Decreasing Functions:**When f'(x) < 0, the function is decreasing.**Stationary Points:**When $f'(x) = 0$, we encounter a stationary point.

**Example: Analysing Function Behaviour**

For the function $y = x^3 + 3x^2 - 9x + 4$, determine the values of $x$ where the graph is increasing or decreasing.

**Solution:**

**1. Finding the Derivative:** $\frac{dy}{dx} = 3x^2 + 6x - 9$

**2. Identifying Stationary Points:**

Solve $3x^2 + 6x - 9 = 0$ to find $x = 1$ and $x = -3$.

**3. Analysing Intervals:**

- The derivative forms a U-shaped parabola.
- The function is increasing for x < -3 and x > 1.
- The function is decreasing for -3 < x < 1

## Determining the Nature of Stationary Points

1. **Find the Second Derivative:**

Compute $\frac{d^2y}{dx^2}$.

**2. Substitute the Stationary Point:**

Substitute the $x$-value of the stationary point into the second derivative.

**3. Interpret the Result:**

- If \frac{d^2y}{dx^2} > 0, it's a minimum point.
- If \frac{d^2y}{dx^2} < 0, it's a maximum point.

**Example: Identifying Stationary Points**

Find the stationary points for $y = x^3 + 3x^2$ and determine their nature.

**Solution:**

**1. First Derivative:** $\frac{dy}{dx} = 3x^2 + 6x$

**2. Stationary Points:**

Set $3x^2 + 6x = 0$ to find $x = -2$ and $x = 0$.

**3. Second Derivative:** $\frac{d^2y}{dx^2} = 6x + 6$

**4. Nature of Stationary Points:**

- For $x = -2$, $\frac{d^2y}{dx^2} = -6$ (Maximum Point at $(-2, 4)$).
- For $x = 0$, $\frac{d^2y}{dx^2} = 6$ (Minimum Point at $(0, 0)$).

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.