• If $\ f'(x) > 0$ for all $\ x$ in an interval, then $\ f(x)$is increasing on that interval.
• If $\ f'(x) < 0$ for all $\ x$ in an interval, then $\ f(x)$ is decreasing on that interval.
• If $\ f'(x) = 0$ for all $\ x$ in an interval, the function is constant on that interval.

• Critical Points: Critical points occur where the derivative $f'(x)$ is zero or undefined. These points are important because they can indicate where the function changes from increasing to decreasing (or vice versa).

• Test Intervals: After finding the critical points, divide the domain of the function into intervals between these points. Determine the sign of the derivative within each interval to see whether the function is increasing or decreasing.

1. Find the Derivative: Calculate the first derivative $\ f'(x)$ of the function $\ f(x).$
2. Find Critical Points: Solve the equation $\ f'(x) = 0$ to find critical points. Also, identify where $\ f'(x)$ is undefined.
3. Determine Sign of $\ f'(x)\ :$

• Use test points in each interval (between critical points) to determine the sign of $\ f'(x)$in that interval.
• If $\ f'(x) > 0$ in an interval, the function is increasing there.
• If $\ f'(x) < 0$ in an interval, the function is decreasing there.

4. Conclude: Based on the sign of the derivative in each interval, determine where the function is increasing or decreasing.

Example 1: Find where the function $\ f(x) = x^3 - 3x^2 + 2$ is increasing or decreasing.

• Step 1: Find the derivative: $f'(x) = 3x^2 - 6x$

• Step 2: Find the critical points:
• Set $\ f'(x) = 0$: $3x^2 - 6x = 0 \quad \Rightarrow \quad 3x(x - 2) = 0$
• So, $\ x = 0$ and $\ x = 2$ are critical points.

• Step 3: Determine the sign of $\ f'(x)$ in the intervals:
• Test intervals: $\ (-\infty, 0) \, \ (0, 2) , \ (2, \infty) .$
• For$\ x \in (-\infty, 0)$, pick $\ x = -1:$ $f'(-1) = 3(-1)^2 - 6(-1) = 3 + 6 = 9 > 0 \quad \text{(:success[increasing])}$
• For $\ x \in (0, 2),$ pick $\ x = 1:$ $f'(1) = 3(1)^2 - 6(1) = 3 - 6 = -3 < 0 \quad \text{(:success[decreasing])}$
• For $x \in (2, \infty$), pick $x = 3$: $f'(3) = 3(3)^2 - 6(3) = 27 - 18 = 9 > 0 \quad \text{(:success[increasing])}$
• Conclusion:
• $\ f(x)$ is increasing on $\ (-\infty, 0)$ and $\ (2, \infty) .$
• $\ f(x)$ is decreasing on $\ (0, 2) .$

Example 2: Determine where the function $\ f(x) = \frac{1}{x}$ is increasing or decreasing.

• Step 1: Find the derivative: $f'(x) = -\frac{1}{x^2}$
• Step 2: Analyse the sign of $\ f'(x):$
• Since $\ f'(x) = -\frac{1}{x^2}$ is always negative for $\ x \neq 0$, the function is decreasing for all $\ x \in (-\infty, 0) \cup (0, \infty).$
• Conclusion:
• $\ f(x) = \frac{1}{x} \ is\ :success[decreasing]\ on\ \ (-\infty, 0) \cup (0, \infty) .$

• Increasing and decreasing functions describe how a function's output changes as the input increases.
• The first derivative $\ f'(x)$ is the key tool in determining whether a function is increasing or decreasing.
• By finding critical points and analysing the sign of the derivative, you can identify intervals where the function increases or decreases, which is essential for understanding the behaviour of the function and solving optimization problems.