Part 5/7:
For instance, in the cubic function ( y = x^3 ), while the derivative equals zero at ( x=0 ), it does not yield a local maximum or minimum, as the function continues to decrease after this point. Despite this, ( x=0 ) remains a critical point; understanding this dichotomy is fundamental in calculus.
The Closed Interval Method
To find absolute extrema, the closed interval method is a vital approach. This method involves a few structured steps:
Identify Critical Points: Find ( c ) such that ( f'(c) = 0 ) or ( f'(c) ) does not exist.
Evaluate Endpoints: Calculate the function's value at both ends of the interval, denoting points ( A ) and ( B ).