A function is defined as increasing on an interval if, for any two numbers and in that interval, implies . Graphically, this means the curve moves upwards or stays flat as it moves from left to right.
A function is strictly increasing if implies , meaning the function never flattens out and always gains value.
Conversely, a function is decreasing if implies , and strictly decreasing if . This corresponds to a downward slope on a Cartesian plane.
The monotonicity of a function refers to whether it is entirely increasing or entirely decreasing over a specific domain.
The First Derivative represents the instantaneous rate of change or the gradient of the tangent to the curve at any point .
If for all in an interval, the function is strictly increasing because the positive gradient indicates the output is growing relative to .
If for all in an interval, the function is strictly decreasing because the negative gradient indicates the output is shrinking as increases.
At points where , the function is stationary. These points often act as the boundaries between increasing and decreasing intervals.
Step 1: Differentiate: Find the first derivative of the given function using standard differentiation rules.
Step 2: Set up Inequalities: To find where the function is increasing, set . To find where it is decreasing, set .
Step 3: Solve the Inequality: Determine the range of values that satisfy the inequality. This often involves solving quadratic or polynomial inequalities.
Step 4: Define Intervals: Express the final answer using interval notation (e.g., ) or inequality notation (e.g., ).
It is vital to distinguish between the value of the function and the value of its derivative . A function can be positive () while it is decreasing ().
| Term | Condition | Visual Description |
|---|---|---|
| Increasing | Moving uphill or flat | |
| Strictly Increasing | Moving uphill only | |
| Decreasing | Moving downhill or flat | |
| Strictly Decreasing | Moving downhill only |
Confusing with : Students often solve (where the graph is above the x-axis) instead of (where the graph is sloping upwards).
Algebraic Errors in Inequalities: When solving , remember that multiplying or dividing an inequality by a negative number flips the inequality sign.
Ignoring the Domain: Some functions have restricted domains (like or ); intervals of increase/decrease must exist within these valid domains.
This concept is the foundation for finding Stationary Points. A point where a function changes from increasing to decreasing is a local maximum, while a change from decreasing to increasing is a local minimum.
In real-world Optimization problems, identifying increasing intervals helps determine when a variable (like profit or velocity) is improving over time.