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 $(0, 0)$).