1 Extreme Values

Why find extreme values of functions? To find where functions are optimized Find maximum profit Find minimum cost Find best dose of medicine Find best gas mileage per speed Etc….

Extrema (plural of extreme) Either minima or maxima Y values Occur at either x=c or at the point (c, f(c)) Absolute or global extrema are the highest or lowest y value for the whole function – there can be at most one for each type Relative or local extrema are a high or low value relative to the values around it – there can be more than one of either type If no interval is specified in the problem, then we assume we are talking about the whole domain of the function Relative extrema CANNOT occur at the endpoints of an interval. Extrema do not have to be nice, tidy hills and valleys, they can be corners or cusps or discontinuities

Extreme Value Theorem (EVT) If f(x) is continuous on a closed interval [a,b], then f(x) has both a max and a min within that interval

So where do these extreme values occur? The theorem doesn’t help us with that! The graph of f(x) is shown below. What is the value of the derivative where the relative min and max occur? If the derivative exists at the extreme value, f’ will always be = 0

Can relative extrema occur at any other place other than where f’(x)=0? Example 2: Sketch the graph of and graphically find any relative extrema.

Critical Values A critical value of a function is a value x=c such that f’(c) = 0 OR f’(c) DNE. Relative extrema can only occur in an open interval at a critical value. Absolute extrema can only occur at a critical value OR at an endpoint of an interval.

Steps for finding absolute extrema Find critical points (f’(c)=0 or does not exist) Evaluate f(x) at each critical point Evaluate f(x) at each endpoint of interval The smallest # from steps 2 and 3 is min The largest # from steps 2 and 3 is max

Example 3 Determine if the EVT applies. If so, find the extrema of on the interval [-1,2].

Example 4 Determine if the EVT applies. If so, find the extrema of on the interval [-8,1].

Example 5 Determine if the EVT applies. If so, find the extrema of on the interval

Example 6 – What do we do if there are not endpoints? Find the extrema of . Use your calculator to verify.

Example 7 Find the extrema of

Example 8 It’s worth pointing out that even though the sufficient conditions of the EVT are met (the “if” part), a function may still have both an absolute max and min.