# Concavity and inflections ## Concave up A function $f$ is **concave up** on an open differentiable interval $I$ if the derivative $f'$ is an increasing function on $I$, then $f'' > 0$. Obtaining tangent line above the graph. ## Concave dowm A function $f$ is **concave down** on an open and differentiable interval $I$ if the derivative is a decreasing function on $I$, then $f'' < 0$. Obtaining tangent lines below the graph. ## Inflection points The function $f$ has an inflection point at $x_0$ if 1. the tangent line in $(x_0, f(x_0))$ exists, and 2. the concavity of $f$ is opposite on opposite sides of $x_0$. If $f$ has an inflection point at $x_0$ and $f''(x_0)$ exists, then $f''(x_0) = 0$ ## The second derivative test ...