Absolute Extrema Extreme Value Theorem in Closed Interval with Examples

#globalmathinstiute #anilkumarmath Test Question:   • Absolute Maximum Minimum Closed Interval M...  
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   • 2x^5/3 - 5x^2/3 [-8, 2] Absolute Maximum M...  
a. Graph a cubic function with absolute minimum at (-2, -10), local maximum at (0, 4), local minimum at (2, -2) and absolute maximum at (4, 11).
b. What is the domain and range of this function?
c. Write the increasing/decreasing interval for the function.
Absolute Maximum and Absolute minimum value for any function continuous in closed interval [a, b] will always exist at the critical numbers or at the end points.
Check the value of the function at the critical numbers and at the end-points to find the result.
Increasing Decreasing Interval Details:    • Test Series Increasing Decreasing Interval...  
Steps to find increasing and decreasing interval of any function, f(x), are:
find the first derivative, f'(x)
find critical numbers, f'(x) = 0 or does not exist (DNE)
INTERVAL TABLE TEST:
These critical numbers divide the domain in intervals. Test each interval with a test point.
RESULT:
If f'(x) is greater than 0 then f(x) is increasing.
If f'(x) is less than 0 then f(x) is decreasing.
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