I'm studying the derivate and monotonicity of functions, and was wondering if such a function exists.
A function whose derivative is negative on $D_f$ but $f$ is not strictly decreasing on $D_f$
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calculus
derivatives
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0Please use the body of the Question to give a self-contained presentation of the problem. In particular a proper statement of the problem would define what $D_f$ means. – 2017-01-15
1 Answers
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If $D_f$ does not need to be connected, the examples are legion: Consider, for example, the domain $D_f = (0,1) \cup(1,2)$ and the map \begin{align*} f \colon D_f &\to \mathbf R \\ x &\mapsto \begin{cases} -x & x < 1\\ 2-x & x > 1 \end{cases} \end{align*}Then $f \in C^1(D_f)$ mit $f'(x) = -1$, $x \in D_f$, but $f(\frac 12) = -\frac 12 \not> \frac 12 = f(\frac 32)$.
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1Perhaps it would be better to use "interval" instead of "connected", as the two are the same in $\Bbb R$ and the latter might confuse some. – 2017-01-15