Concavity describes the way a graph bends, not just whether it is going up or down. The second derivative, written f''(x), measures how the slope f'(x) is changing as x changes. This matters because two graphs can both be increasing while bending in very different ways.
Concavity helps you sketch functions, identify turning behavior, and understand motion more deeply.
When f''(x) > 0, the slopes are increasing and the graph is concave up, like a cup. When f''(x) < 0, the slopes are decreasing and the graph is concave down, like a cap. A point where concavity changes is called an inflection point, and it often occurs where f''(x) = 0 or where f''(x) is undefined.
In motion problems, the second derivative of position is acceleration, so concavity shows whether velocity is increasing or decreasing.
Key Facts
- The second derivative is the derivative of the first derivative: f''(x) = d/dx[f'(x)].
- If f''(x) > 0 on an interval, then f is concave up on that interval.
- If f''(x) < 0 on an interval, then f is concave down on that interval.
- An inflection point occurs where the graph changes concavity.
- For position s(t), velocity is v(t) = s'(t) and acceleration is a(t) = s''(t).
- A possible inflection point can occur where f''(x) = 0 or where f''(x) does not exist, but concavity must actually change.
Vocabulary
- Concavity
- Concavity describes whether a graph bends upward like a cup or downward like a cap.
- Second derivative
- The second derivative f''(x) measures how the slope of a function is changing.
- Concave up
- A graph is concave up on an interval when its slopes are increasing and f''(x) is positive.
- Concave down
- A graph is concave down on an interval when its slopes are decreasing and f''(x) is negative.
- Inflection point
- An inflection point is a point on a graph where the concavity changes from up to down or from down to up.
Common Mistakes to Avoid
- Confusing increasing with concave up. A function can be increasing while concave down if its slopes are positive but getting smaller.
- Assuming f''(x) = 0 always means an inflection point. This is wrong because concavity must change on the two sides of the point.
- Using f'(x) instead of f''(x) to decide concavity. The first derivative tells whether the function is increasing or decreasing, while the second derivative tells how the slope is changing.
- Forgetting to test intervals around possible inflection points. You must check the sign of f''(x) on each side to confirm where the graph is concave up or concave down.
Practice Questions
- 1 For f(x) = x^3 - 6x^2 + 9x + 1, find f''(x), determine where the graph is concave up or concave down, and identify any inflection point.
- 2 For s(t) = 2t^3 - 15t^2 + 24t, find the acceleration a(t), then determine when the motion changes from concave down to concave up.
- 3 A graph is increasing on an interval, but its tangent slopes are getting smaller as x increases. Explain whether the graph is concave up or concave down and justify your answer using the second derivative.