22 lines
745 B
Markdown
22 lines
745 B
Markdown
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# Concavity and inflections
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## Concave up
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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.
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## Concave dowm
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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.
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## Inflection points
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The function $f$ has an inflection point at $x_0$ if
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1. the tangent line in $(x_0, f(x_0))$ exists, and
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2. the concavity of $f$ is opposite on opposite sides of $x_0$.
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If $f$ has an inflection point at $x_0$ and $f''(x_0)$ exists, then $f''(x_0) = 0$
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## The second derivative test
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