A derivative graph shows how the original function is changing at each x-value. When you read the graph of f'(x), you are not looking at the height of f(x), but at the slope of f(x). This matters because slope information tells you where the original function rises, falls, turns around, and changes curvature.
A single graph of f'(x) can reveal the main shape of f(x) without ever seeing f(x) directly.
The key idea is to compare f'(x) with the x-axis. Where f'(x) is positive, f(x) is increasing, and where f'(x) is negative, f(x) is decreasing. When f'(x) crosses zero, f(x) may have a local maximum or local minimum, depending on how the sign changes.
Inflection points of f(x) occur where f'(x) changes from increasing to decreasing or from decreasing to increasing, which means at local extrema of f'(x).
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.
- If f'(c) = 0 and f'(x) changes from positive to negative at c, then f(x) has a local maximum at x = c.
- If f'(c) = 0 and f'(x) changes from negative to positive at c, then f(x) has a local minimum at x = c.
- If f'(x) changes from increasing to decreasing or decreasing to increasing at x = c, then f(x) may have an inflection point at x = c.
- f''(x) is the slope of f'(x), so f''(x) > 0 means f(x) is concave up and f''(x) < 0 means f(x) is concave down.
Vocabulary
- Derivative
- The derivative f'(x) gives the instantaneous rate of change or slope of the original function f(x).
- Increasing interval
- An interval where f(x) rises as x increases, which happens when f'(x) is positive.
- Critical point
- A point in the domain of f where f'(x) = 0 or f'(x) does not exist.
- Local extremum
- A local maximum or local minimum where the function changes from increasing to decreasing or decreasing to increasing.
- Inflection point
- A point where the concavity of f(x) changes from up to down or from down to up.
Common Mistakes to Avoid
- Reading the height of f'(x) as the height of f(x). The value of f'(x) tells the slope of f(x), not the y-value of f(x).
- Assuming every zero of f'(x) is a maximum or minimum. A local extremum occurs only if f'(x) changes sign at that zero.
- Confusing extrema of f'(x) with extrema of f(x). Peaks and valleys on the derivative graph usually indicate possible inflection points of f(x), not turning points of f(x).
- Ignoring intervals and checking only single points. Whether f(x) increases, decreases, or changes concavity depends on the sign or trend of f'(x) over an interval.
Practice Questions
- 1 The graph of f'(x) is positive on (-4, -1), negative on (-1, 3), and positive on (3, 6). On which intervals is f(x) increasing and decreasing?
- 2 Suppose f'(x) crosses the x-axis at x = -2 from negative to positive, and at x = 5 from positive to negative. Classify the local extrema of f(x) at x = -2 and x = 5.
- 3 A graph of f'(x) has a local maximum at x = 1 and a local minimum at x = 4, with no breaks in the graph. Explain what these points suggest about the concavity and possible inflection points of f(x).