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docs/mathematics/calculus/concavity-and-inflections.md

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+# 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
+
+...