Mon Prof Augmenté
Bending Along a Curve
Age 17 · Maths
A function is convex where its curve bends upward like a bowl, and its second derivative is positive there. At an inflection point the bend switches between upward and downward, and the tangent line crosses the curve.
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In the picture
- Upward-bending segment: the curve bends like a bowl and its slope increases from left to right.
- Downward-bending segment: the curve bends like a dome and its slope decreases from left to right.
- Change point: the curve switches from upward-bending to downward-bending.
- Tangent line at the change point: it crosses from one side of the curve to the other.
Examples
- A skateboard ramp that curves upward near the bottom and downward near the top.
- A hill road whose slope increases on one side and decreases on the other.
Key takeaways
- Where the second derivative is positive, the curve bends upward and is convex.
- An inflection point is where the curve switches between convex and concave.
Quiz: Bending Along a Curve (Age 17)
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What does a positive second derivative tell you about the curve at that point?
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What happens at an inflection point?
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On an upward-bending segment, what happens to the slope as you move from left to right?
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