# Absolute or global minimum value of f if f x f c for

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Chapter 2 / Exercise 66
Elementary Algebra
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absolute (or global) minimum value of f if f ( x ) f ( c ) for all x D . Definition 23. Let c lie in the domain D of the function f . Then f ( c ) is a local maximum value of f if f ( x ) f ( c ) for all x D that lie in some open interval containing c . f ( c ) is a local minimum value of f if f ( x ) f ( c ) for all x D that lie in some open interval containing c . Maxima and minima of a function f are called extreme values , or simply extrema of f . Example 51. f ( x ) = x 2 has an absolute (and local) minimum value of 0 at x = 0 and no (absolute of local) maxima, whereas g ( x ) = 1 - 4 x 2 has an absolute (and local) maximum value of 1 at x = 1 and no (absolute of local) minima. Some functions possess extreme values while others do not. Continuous functions always attain both absolute maxima and minima on closed intervals. This is the content of the following result. Theorem 31 (The Extreme Value Theorem) . If f is continuous on a closed interval [ a, b ] , then f attains both an absolute maximum as well as an absolute minimum on [ a, b ] . Both of the conditions (continuity and the fact that the interval in question is closed) are necessary as illustrated by the following example. 51
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Chapter 2 / Exercise 66
Elementary Algebra
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52 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION Example 52. f ( x ) = 1 x does not attain an absolute maximum value on the interval (0 , 1), even though it is continuous on this interval. The function g ( x ) = ( x 2 + 1 if x < 1 2 - x if x 1 fails to attain an absolute maximum on the interval [0 , 2], even though this interval is closed. To find the absolute maximum and minimum values of a continuous function on a closed interval, we will need to first find the local extrema for the function on the interval. We therefore turn to determining how to do this. Theorem 32 (Fermat’s Theorem) . If f has a local extremum at c , then either f 0 ( c ) fails to exist or f 0 ( c ) = 0 . Note: This is only a one directional implication. It states that the only possible places where a function could have a local extremum are at the points where the derivative is zero or doesn’t exist. There is no guarantee that these points will actually correspond to extreme values. Also, it is important to check where the derivative fails to exist in addition to where it is equal to zero. We now illustrate these two points in the following example. Example 53. The function f given by f ( x ) = x 3 has a vanishing derivative at x = 0, but fails to have a local extremum at this point. Further, the functions g and h given by g ( x ) = | x | and h ( x ) = 3 x 2 both have a global (and therefore local) minimum at x = 0 even though neither is differentiable at this point. Having seen that the values of x at which a function either has vanishing derivative or fails to be differentiable are the only possible places where a function can attain a local extremum, we now give these points a special name.
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