Increasing and decreasing functions describe how a graph moves as you read it from left to right. In calculus, the first derivative gives a precise way to identify this behavior without relying only on a sketch. If f'(x) is positive on an interval, the function is increasing there, and if f'(x) is negative, the function is decreasing there.
This idea helps students connect the shape of a graph to an algebraic test.
Understanding Calculus: Increasing and Decreasing Functions
The derivative measures the function's instantaneous rate of change. On a graph, it describes the slope of the tangent line at a point. A steep positive slope means output values are rising quickly near that input.
A shallow negative slope means they are falling slowly. A horizontal tangent has slope zero, but this alone does not prove that the graph turns around.
For example, the graph of x cubed has a horizontal tangent at zero and keeps moving upward through that point. The important evidence is the behavior on each side of the point, not only the slope at the point itself.
A reliable method begins with the domain of the original function. Next, find its derivative and locate every input where the derivative is zero. Include places where the derivative is undefined only when the original function has a value there.
These inputs split the number line into intervals. Choose one test value inside each interval and determine the derivative's sign there. One test value works because the sign cannot switch within an interval unless the derivative reaches zero or fails to exist.
Factoring the derivative often makes this faster. Each factor contributes a positive or negative sign, so students can track signs without doing large calculations repeatedly.
Some critical numbers need extra care. A factor with an even power usually keeps the same sign on both sides of its zero. In that case, the graph may flatten without changing direction.
A factor with an odd power usually changes sign as the input passes through its zero. Fractions require attention to both numerator and denominator. A denominator of zero can create a break in the graph rather than a turning point.
That input is not part of the function's domain, so it cannot be a local maximum or local minimum. Endpoints matter too. On a closed interval, the largest or smallest value may occur at an endpoint, even though no sign change happens there.
This test appears in problems about motion, growth, business, and science. If a position function has a positive velocity, an object moves in the chosen positive direction. If velocity is negative, it moves in the opposite direction.
For a model of temperature, population, or profit, the derivative tells whether the quantity is currently rising or falling. Units help make the result meaningful. If height is measured in meters and time in seconds, the derivative is measured in meters per second.
When learning this topic, keep the function, its derivative, and the sign chart separate. Many errors come from solving the derivative correctly but testing the wrong intervals, forgetting domain restrictions, or confusing a zero derivative with a guaranteed maximum or minimum.
Key Facts
- If f'(x) > 0 on an interval, then f(x) is increasing on that interval.
- If f'(x) < 0 on an interval, then f(x) is decreasing on that interval.
- Critical numbers occur where f'(x) = 0 or where f'(x) does not exist, as long as f(x) is defined.
- A sign chart tests the sign of f'(x) on intervals split by critical numbers.
- If f'(x) changes from positive to negative at c, then f has a local maximum at x = c.
- If f'(x) changes from negative to positive at c, then f has a local minimum at x = c.
Vocabulary
- Increasing function
- A function is increasing on an interval if its output values rise as x moves from left to right.
- Decreasing function
- A function is decreasing on an interval if its output values fall as x moves from left to right.
- First derivative
- The first derivative f'(x) gives the instantaneous rate of change or slope of the tangent line to f(x).
- Critical number
- A critical number is an x-value in the domain of f where f'(x) = 0 or f'(x) does not exist.
- Local extremum
- A local extremum is a local maximum or local minimum where a function is higher or lower than nearby values.
Common Mistakes to Avoid
- Using f(x) instead of f'(x) to decide increasing or decreasing is wrong because the sign of the function value does not tell the direction of change.
- Assuming every critical number is a local maximum or minimum is wrong because the derivative may not change sign there.
- Testing only the critical number in a sign chart is wrong because f'(x) may be zero or undefined at that point, so you must test points inside each interval.
- Forgetting domain restrictions is wrong because intervals of increase and decrease must be stated only where the original function is defined.
Practice Questions
- 1 For f(x) = x^2 - 4x + 1, find f'(x), the critical number, and the intervals where f is increasing and decreasing.
- 2 For f(x) = x^3 - 3x^2 - 9x + 5, use a sign chart for f'(x) to find the intervals of increase and decrease and identify any local extrema.
- 3 A function has f'(x) < 0 on (-infinity, 2), f'(2) = 0, and f'(x) > 0 on (2, infinity). Explain what happens to the graph at x = 2 and why.