Convexity of a Graph & Inflection Points
Big Idea: The second derivative tells us about the shape of a curve — whether it bends upward like a bowl or downward like a hill. The points where this bending changes direction are called inflection points.
1. Concavity (Convexity) of a Graph
When we look at a function's graph, we can observe two types of curvature:
🥣 Concave Up (Convex)
The graph curves upward, like the inside of a bowl.
- The tangent line lies below the curve
- The slope $f'(x)$ is increasing
- $f''(x) > 0$ on the interval
🏔️ Concave Down (Concave)
The graph curves downward, like the top of a hill.
- The tangent line lies above the curve
- The slope $f'(x)$ is decreasing
- $f''(x) < 0$ on the interval
📐 Formal Definitions
Definition 1: A function $f$ is concave up on an interval $I$ if for all $x \in I$:
$$f''(x) > 0$$
Definition 2: A function $f$ is concave down on an interval $I$ if for all $x \in I$:
$$f''(x) < 0$$