Significance Of The Derivative

Defintions:

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let $f$ be a function and $A$ a set of numbers contained in the domain of $f$. A point $x$ in $A$ is a maximum point for $f$ on $A$ if $f(x) \ge f(y)$ for every $y$ in $A$. The number $f(x)$ itself is called the maximum value of $f$ on $A$ (and we also say that $f$ "has its maximum value on $A$ at $x$")