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## 5.2: Relative Extrema

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**5.2: Relative Extrema**Objective: To use the derivative to find maximum and minimum values of a function**Discuss what is happening in this graph. Are there any**maximum or minimum points?**RELATIVE MAXIMUM OR MINIMUM**• Let c be a number in the domain of f. • f(c) is a relative (or local) maximum if on the open interval (a,b)f(x) < f(c) for all x in (a,b) (f(c) is the largest value) • f(c) is a relative (or local) minimum if on the open interval (a,b)f(x) > f(c) for all x in (a,b) (f(c) is the smallest value) • A function has a relative (or local) extremum (plural extrema) if at c there is either a local max or min**Relative Extrema at Endpoints**• If c is an endpoint of the domain of f, consider what is happening to the left or right of endpoint: • If c is a left-hand endpoint: • Relative max if f is decreasing to the right of c • Relative min if f is increasing to the right of c • If c is a right-hand endpoint: • Relative max if f is increasing to the left • Relative min if f is decreasing to the left**Identify the x-values of all the points where the graph has**relative extrema:**If a function has a relative extremum at c, then c is a**critical number or c is an endpoint of the domain…This is a big deal!!!! • This does not mean that all critical numbers are relative extrema →**FIRST DERIVATIVE TEST!!!**• Find critical numbers, c • Test the sign of f’(x) to the left and the right of critical numbers • f(c) is a relative max if: • f’(x) > 0 to the left of c • f’(x) < 0 to the right of c • f(c) is a relative min if: • f’(x) < 0 to the left of c • f’(x) > 0 to the right of c • For endpoints, it depends if it is a left or right-hand endpoint • If no sign change, no relative extreme values at c**Find the x value of all points where the function has**relative extrema. Find the value(s) of any relative extrema.