Monotonicity describes whether a function keeps increasing, keeps decreasing, or stays constant as x moves from left to right. The derivative is the main tool for testing this behavior because it measures the slope of the tangent line at each point. When the derivative is positive on an interval, the graph rises, and when it is negative, the graph falls.
This matters because monotonic functions are easier to analyze, graph, and use in applications such as motion, optimization, and modeling.
The sign of f'(x) gives a precise way to prove monotonicity, not just guess from a picture. Critical points, where f'(x) = 0 or f'(x) is undefined, often divide the number line into intervals where the derivative sign can be tested. If a function is strictly increasing or strictly decreasing on its whole domain, then it is one-to-one and can have an inverse function.
This connection is important because many inverse relationships in science and mathematics depend on a function passing the horizontal line test.
Key Facts
- 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 for all x in an interval, then f is constant on that interval.
- Critical points occur where f'(x) = 0 or where f'(x) does not exist.
- A function that is strictly increasing or strictly decreasing on its domain is one-to-one.
- If f is one-to-one and differentiable with f'(a) != 0, then (f^-1)'(f(a)) = 1 / f'(a).
Vocabulary
- Monotonic function
- A function that consistently increases, consistently decreases, or remains constant on an interval.
- Increasing interval
- An interval where larger x-values correspond to larger function values.
- Decreasing interval
- An interval where larger x-values correspond to smaller function values.
- Derivative sign chart
- A number line diagram that uses the sign of f'(x) on intervals to determine where a function increases or decreases.
- One-to-one function
- A function in which each output value comes from at most one input value.
Common Mistakes to Avoid
- Confusing f(x) > 0 with f'(x) > 0 is wrong because a function can be above the x-axis while still decreasing.
- Testing only one point without finding critical points is wrong because the derivative sign can change only after zeros or undefined points of f'(x).
- Assuming f'(x) = 0 at one point means the function is constant is wrong because a single horizontal tangent does not control the whole interval.
- Claiming every increasing-looking graph is one-to-one without checking the whole domain is wrong because the function might turn around outside the visible window.
Practice Questions
- 1 For f(x) = x^2 - 6x + 5, find f'(x), determine where f is increasing and decreasing, and identify the critical point.
- 2 For g(x) = x^3 - 3x^2 - 9x + 4, use g'(x) to make a sign chart and find the intervals where g is increasing and decreasing.
- 3 A differentiable function has f'(x) > 0 for all real x. Explain why the function must be one-to-one and why this makes an inverse function possible.