Explanation: Local Extrema on a Closed Interval

(related to Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum)

The proposition about the zero-derivative as a necessary condition for a local extremum can be used to test if a value $x$ can be a local extremum. If $f'(x)\neq 0$ for some $x$, then we have a proof that $x$ cannot be an extremum. But if $f'(x)=0$, $x$ can be a local extremum (but does not have to, taking the function $f(x):=x^3$ as an example).


When using this test, it is important to make sure that the function $f$ is defined on an open real interval $]a,b[$. For if the interval was closed, then our test might fail. Take the function $f:[0,1]\to\mathbb R$, $x\to x$ as an example. It has a minimum at $0$ and a maximum at $1$, but it has nevertheless a non-zero derivative $f'=1$. Thus, according to a corresponding proposition, every continuous function on a closed interval takes its maximum (or minimum), even though its derivative (if it exists) might be non-zero.

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983