Theorem 2 says that if a function has a first derivative at an interior point where there is a local extremum, then the derivative must equal zero at that point. Theorem 2 If a function has a local maximum value or a local minimum value at an interior point c of its domain and if f ' exists at c, then f ' ( c) = 0. Theorem 2 below, which is also called Fermat's Theorem, identifies candidates for local extreme-value points. So, absolute extrema can be found by investigating all local extrema.Ĭandidates for Local Extreme-Value Points It is clear from the definitions that for domains consisting of one or more intervals, any absolute extreme point must also be a local extreme point. The definition can be extended to include endpoints of intervals. The definition of local extrema given above restricts the input value to an interior point of the domain. local minimum at c if and only if f( c)į( x) for all x in some open interval containing c.local maximum at c if and only if f( x)į( c) for all x in some open interval containing c.Let c be an interior point of the domain of the function f. Local extreme values, as defined below, are the maximum and minimum points (if there are any) when the domain is restricted to a small neighborhood of input values. One of the most useful results of calculus is that the absolute extreme values of a function must come from a list of local extreme values, and those values are easily found using the first derivative of the function. It does not address how to find the extreme values. This theorem says that a continuous function that is defined on a closed interval must have both an absolute maximum value and an absolute minimum value. Theorem 1 If f is continuous on a closed interval, then f has both an absolute maximum value and an absolute minimum value on the interval. The theorem is important because it can guide our investigations when we search for absolute extreme values of a function. It describes a condition that ensures a function has both an absolute minimum and an absolute maximum. Theorem 1 below is called the Extreme Value theorem. The two examples above show that the existence of absolute maxima and minima depends on the domain of the function. As shown below, the graph on the interval suggests that f has an absolute maximum of 9 at x = 3 and an absolute minimum of 0 at x = 0. If the domain of f( x) = x 2 is restricted to, the corresponding range is. The Absolute Extreme Values on a Restricted Domain The graph in the figure below suggests that the function has no absolute maximum value and has an absolute minimum of 0, which occurs at x = 0. The domain of f( x) = x 2 is all real numbers and the range is all nonnegative real numbers. absolute minimum value of f on D if and only if f( c).absolute maximum value of f on D if and only if f( x).Let f be a function with domain D and let c be a fixed constant in D. When an output value of a function is a maximum or a minimum over the entire domain of the function, the value is called the absolute maximum or the absolute minimum, as defined below. Both absolute and local maximum and minimum values are of interest in many contexts. In such problems there may be a largest or smallest output value over the entire input interval of interest or within a local neighborhood of an input value. Optimization problems are one of the most important applications of differential calculus because we often want to know when the output of a function is at its maximum or minimum.
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