Increasing and Decreasing Functions
One of the most powerful applications of the derivative is determining where a function is increasing or decreasing — a property collectively called monotonicity. A function is increasing on an interval if its output values rise as the input moves to the right, and decreasing if its output values fall. Rather than checking infinitely many points, calculus gives us a systematic shortcut: examine the sign of the first derivative f′(x). The connection between f′(x) and monotonicity stems from the derivative's geometric meaning. Because f′(x) represents the slope of the tangent line at each point, a positive slope means the curve is climbing, a negative slope means it is falling, and a zero slope signals a potential turning point. By solving the inequality f′(x) > 0 or f′(x) < 0 over appropriate intervals, we can produce a complete monotonicity analysis without ever plotting the graph. Monotonicity analysis is an essential building block for curve sketching, optimization, and understanding real-world rates of change. Whether you are modelling the height of a projectile, the profit of a company, or the spread of a disease, knowing where the underlying function increases or decreases reveals critical information about the behavior of the system you are studying.
- Find the First Derivative
- Find Critical Numbers
- Build a Sign Chart
- Apply the Monotonicity Test
- State the Result Precisely
Find the First Derivative
Differentiate f(x) using standard rules (power, product, quotient, chain). The result, f′(x), gives the instantaneous rate of change at every x in the domain.
Find Critical Numbers
Set f′(x) = 0 and solve for x. Also identify any x values where f′(x) is undefined but f(x) itself is defined. These are the critical numbers; they are the only places where the function can switch between increasing and decreasing.
Build a Sign Chart
Place the critical numbers on a number line. They divide the real line into open intervals. Choose one test value inside each interval and evaluate the sign of f′ at that test value — you only need the sign, not the exact value.
Apply the Monotonicity Test
If f′(x) > 0 for all x in an interval, then f is increasing on that interval. If f′(x) < 0 for all x in an interval, then f is decreasing on that interval. If f′(x) = 0 throughout an interval, f is constant there. Record these conclusions carefully, using interval notation.
State the Result Precisely
Write your answer in the form: 'f is increasing on (a, b) and decreasing on (b, c).' Open intervals are conventionally used because the derivative test says nothing about the behavior exactly at the endpoints of each piece. A local maximum occurs where f changes from increasing to decreasing, and a local minimum where it changes from decreasing to increasing.
The Key Equation
- f(x) — Original function (varies): The function whose monotonicity we are analyzing.
- f'(x) — First derivative (varies): The instantaneous rate of change of f with respect to x; its sign determines whether f rises or falls.
- I — Interval (dimensionless): An open interval on the x-axis, typically bounded by consecutive critical numbers.
- \forall x \in I — For all x in I (—): Logical quantifier meaning the condition must hold at every point in the interval, not just at the test value.
Worked Example
Problem. Find the intervals on which f(x) = x³ − 6x² + 9x + 2 is increasing and decreasing.
Answer. f is increasing on (−∞, 1) and (3, +∞); decreasing on (1, 3). Local max at x = 1, local min at x = 3.
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
Leibniz independently developed calculus alongside Isaac Newton, and crucially invented much of the notation we still use today — including dy/dx for the derivative and the integral sign ∫. His clear symbolic framework made it straightforward to express conditions like f′(x) > 0 and reason about increasing or decreasing functions algebraically. Leibniz designed one of the first mechanical calculators, the Stepped Reckoner, which could multiply and divide — a remarkable feat of engineering in the 1670s.
Related Concepts
- Critical Points — Critical numbers are found by setting f′(x) = 0 and are the boundaries between increasing and decreasing intervals.
- Local Extrema and the First Derivative Test — Once increasing and decreasing intervals are identified, local maxima and minima are classified at each critical point.
- Concavity and the Second Derivative — The second derivative f″(x) extends the analysis by revealing whether f′ itself is increasing or decreasing, describing the shape of the curve.
- Curve Sketching — Monotonicity is one of several ingredients — along with intercepts, concavity, and asymptotes — used to sketch accurate graphs of functions.