![]() Have an absolute maximum and an absolute minimum on $$. If a function is continuous on a closed interval, then it achieves both an absolute maximum and an absolute minimum on the interval. The extreme value occurs either at a critical point or at an endpoint. If $f$ has an extreme value on a closed interval, then.The extreme value occurs at a critical point of $f$. If $f$ has an extreme value on an open interval, then. ![]() If $f(x_0) \leq f(x)$ for all x in an interval $I$, then $f$ achieves itsĪbsolute maxima and absolute minima are often refered to simply as maxima and minima and are collectively called extreme values of $f$. If $f(x_0) \geq f(x)$ for all $x$ in an interval $I$, then $f$ achieves its In summary, relative extrema occur where $f'(x)$ changes sign. ![]() $f$ does not have a relative extremum at $x_0$. Left from $x_0$ and an open interval extending right from $x_0$, then To avoid confusion, we ignore most of the subscripts here. Example 59 ended with the recognition that each of the given functions was actually a composition of functions. Let’s go ahead and finish this example out. Use the Chain Rule to find the derivatives of the following functions, as given in Example 59. If $f'(x)$ has the same sign on both an open interval extending In many functions we will be using the chain rule more than once so don’t get excited about this when it happens.$f'(x) 0$ on an open interval extending right from $x_0$, then $f$ If $f'(x) > 0$ on an open interval extending left from $x_0$ and.Suppose $f$ is continuous at a critical point $x_0$. Relative extrema of $f$ occur at critical points of $f$, values Relative maxima and minima are called relative extrema. The function has a relative, or local minimum at $x_0$ if $f(x_0) \leq f(x)$ for all $x$ in some open interval containing $x_0$. A function $f$ has a relative (or local) maximum at $x_0$ if $f(x_0) \geq f(x)$ for all $x$ in some open interval containing $x_0$.
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