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.
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.