Calculus AB: Applications of the Derivative

Problem : On the graph below, find the local and absolute extrema on the interval [a, b]

Solution for Problem 1 >>

The critical points occur where the tangent is horizontal, at points c , d , and e listed below. c and e are local minima. d is a local maximum. The endpoint b is the absolute maximum, and the critical point e is the absolute minimum in this interval.

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Problem : Find the critical points of f (x) = x ³ + x ²

Problem : Does f (x) = x ³ + x ² have an absolute maximum?

Problem : The Extreme Value Theorem doesn't apply to continuous functions on open intervals, but could such a function have both an absolute minimum and an absolute maximum on that open interval?

Solution for Problem 4 >>

Yes, it certainly could. These extrema could occur at critical point on the interior of the interval:

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Problem : Do absolute extrema always count as local extrema?