Changes

First derivative test

, 23:17, 3 May 2012
What the test says: one-sided sign versions
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! Continuity and differentiability assumption !!Hypothesis on sign of derivative!! Conclusion !! Prototypical pictures (the dotted point corresponds to $(c,f(c))$, and the dashed line is the one-sided tangent line at the point)
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| $f$ is left continuous ''at'' $c$ and differentiable on the immediate left of $c$ || $\! f'(x)$ is positive (respectively, nonnegative) for $x$ to the immediate left of $c$ (i.e., for $x \in (c - \delta, c)$ for sufficiently small $\delta > 0$)|| $f$ has a strict local maximum from the left at $c$, i.e., $f(c) > f(x)$ (respectively, $f$ has a local maximum from the left at $c$, i.e., $f(c) \ge f(x)$) for $x$ to the immediate left of $c$. || [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|100px]]
|$f$ is left continuous ''at'' $c$ and differentiable on the immediate left of $c$ ||$\! f'(x)$ is negative (respectively, nonpositive) for $x$ to the immediate left of $c$ (i.e., for $x \in (c - \delta, c)$ for sufficiently small $\delta > 0$)|| $f$ has a strict local minimum from the left at $c$, i.e., $f(c) < f(x)$ (respectively, $f$ has a local minimum from the left at $c$, i.e., $f(c) \le f(x)$) for $x$ to the immediate left of $c$. || [[File:Leftdecreasingconcaveupflat.png|100px]][[File:Leftdecreasingconcavedown.png|100px]]
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| $f$ is right continuous ''at'' $c$ and differentiable on the immediate right of $c$ ||$\! f'(x)$ is positive (respectively, nonnegative) for $x$ to the immediate right of $c$ (i.e., for $x \in (c,c + \delta)$ for sufficiently small $\delta > 0$)|| $f$ has a strict local minimum from the right at $c$, i.e., $f(c) < f(x)$ (respectively, $f$ has a local minimum from the right at $c$, i.e., $f(c) \le f(x)$) for $x$ to the immediate right of $c$. || [[File:Rightincreasingconcaveupnotflat.png|100px]]
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| $f$ is right continuous ''at'' $c$ and differentiable on the immediate right of $c$ ||$\! f'(x)$ is negative (respectively, nonpositive) for $x$ to the immediate right of $c$ (i.e., for $x \in (c,c + \delta)$ for sufficiently small $\delta > 0$)|| $f$ has a strict local maximum from the right at $c$, i.e., $f(c) > f(x)$ (respectively, $f$ has a local maximum from the right at $c$, i.e., $f(c) \ge f(x)$) for $x$ to the immediate right of $c$. ||[[File:Rightdecreasingconcavedownnotflat.png|100px]]
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