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Finding the Absolute Extrema Find all critical numbers of f within the interval [a, b]. Plug in each critical number from step 1 into the function f(x). Plug in the endpoints, a and b, into the function f(x). The largest value is the absolute maximum, and the smallest value is the absolute minimum.
Set the derivative equal to 0 and solve for x. This gives you the x-values of the maximum and minimum points. Plug those x-values back into the function to find the corresponding y-values. This will give you your maximum and minimum points of the function.
Absolute maxima: A point x = a is a point of global maximum for f(x) if f(x) f(a) for all xD (the domain of f(x)). Absolute minima: A point x = a is a point of global minimum for f(x) if f(x) f(a) for all xD (the domain of f(x)).
We say that f(x) has a relative (or local) maximum at x=c if f(x)f(c) f ( x ) f ( c ) for every x in some open interval around x=c . We say that f(x) has an absolute (or global) minimum at x=c if f(x)f(c) f ( x ) f ( c ) for every x in the domain we are working on.
Generally, we can represent the absolute value function as, f(x) = a |x - h| + k, where a represents how far the graph stretches vertically, h represents the horizontal shift and k represents the vertical shift from the graph of f(x) = |x|.
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To find the absolute maximum and minimum values of f on D , do the following: Determine the critical points of f on D . Calculate f at each of these critical points. Determine the maximum and minimum values of f on the boundary of its domain.
How do you find the absolute maximum and minimum values of a continuous function? Calculate the derivative of the function. Identify the critical points by finding where the derivative is equal to zero (or infinity). Wherever the derivative is equal to zero, you will have max, min, or inflection points.
Here we have the following conditions to identify the local maximum and minimum from the second derivative test. x = k, is a point of local maxima if f(k) = 0, and f(k) 0. x = k is a point of local minima if f(k) = 0, and f(k) 0 . The test fails if f(k) = 0, and f(k) = 0.

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