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Objectives

  • Define absolute extrema.
  • Define local extrema.
  • Explain how to find the critical points of a function over a closed interval.
  • Describe how to use critical points to locate absolute extrema over a closed interval.

Summary

The critical numbers of a continuous function f are the values of p for which f ′ ( p ) = 0 or f ′ ( p ) does not exist. These values are important because they identify horizontal tangent lines or corner points on the graph, which are the only possible locations at which a local maximum or local minimum can occur.

Given a differentiable function f , whenever f ′ is positive, f is increasing; whenever f ′ is negative, f is decreasing. The first derivative test tells us that at any point where f changes from increasing to decreasing, f has a local maximum, while conversely at any point where f changes from decreasing to increasing f has a local minimum.

Given a twice differentiable function f , if we have a horizontal tangent line at x = p and f ″ ( p ) is nonzero, the sign of f ″ tells us the concavity of f and hence whether f has a maximum or minimum at x = p . In particular, if f ′ ( p ) = 0 and f ″ ( p ) < 0 , then f is concave down at p and f has a local maximum there, while if f ′ ( p ) = 0 and f ″ ( p ) > 0 , then f has a local minimum at p . If f ′ ( p ) = 0 and f ″ ( p ) = 0 , then the second derivative does not tell us whether f has a local extreme at p or not.

To find relative extreme values of a function, we use a first derivative sign chart and classify all of the function's critical numbers. If instead we are interested in absolute extreme values, we first decide whether we are considering the entire domain of the function or a particular interval.

In the case of finding global extremes over the function's entire domain, we again use a first or second derivative sign chart. If we are working to find absolute extremes on a restricted interval, then we first identify all critical numbers of the function that lie in the interval.

For a continuous function on a closed, bounded interval, the only possible points at which absolute extreme values occur are the critical numbers and the endpoints. Thus, we simply evaluate the function at each endpoint and each critical number in the interval, and compare the results to decide which is largest (the absolute maximum) and which is smallest (the absolute minimum).

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