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Maximum-Minimum Theorem : Continuity and Bounded Functions
continuous on (a,b) such that f(x) not equal 0 for all x element of (a,b) but 1/f(x) is *not* bounded on (a,b)
Note that this is NOT a contradiction of the concluson in part a above because we now have an open, not a closed interval.
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Continuous functions on closed intervals
32645 Continuous functions on closed intervals See attached, ps explain correct answer
Let f be a function that is continuous on the closed interval [0,1] and differentiable on the open interval (0,1).
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Finding Critical Numbers; Finding the Extrema in the Interval; etc.
Rolle's theorem: Let f be differentiable on the open interval (a, b) and continuous on the closed interval [a, b]. Then if , then there is at least one point where .
is not defined at x=-1, hence it is not differentiable in the interval.
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Key data regarding differentiation
a) The maximum of a function that is continuous on a closed interval can occur at two different values in the interval.
b) If a function is continious on a closed interval, then it must have a minimum on the interval.
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Solving Inequalities, Limits and Derivatives
The Max-Min theorem for continuous function states If f is a continuous function on a compact set K, then f has an absolute maximum and an absolute minimum on K.
So it is consistent with the theorem.
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Maximum-Minimum Theorem, Limits, Continuity and Function Composition
Note:
U: means union
Max-min theorem: f: [a, b] ->R continuous on [a, b]. Then f has an absolute max and an absolute min on [a, b]
Solution. For a function f: A -> R, the conclusion of the maximum-minimum theorem does NOT hold.
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Two Segment Graph : Equation of Tangent and Calculation of Points
Moreover, So, we have
(1) which gives rise to g has a relative maximum at x=2;
(2) which gives rise to g has a relative minimum at x=4
So, there is only one point x=2 on the open interval (-2,5) at which g has a relative maximum.
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Differentiation, Extrema of a Function
Graph a function on the interval [-2,5] having the following characteristics:
Critical number at x = 0, but no extrema
Absolute minimum at x = 5
Absolute maximum at x = 2
I assume you have studied derivatives.
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An open-top box is to be made as follows...
Use calculus to find the highest and lowest values attained by the function f on
the interval 0 (greater than or equal to) x (less than or equal to) 5.
As can be found from part a, is a local minimum for . And it is the only local minimum.