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2023-09-23 12:46:18 +02:00
# 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
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