Local Extreme Points

Objectives Students will be able to Find relative maximum and minimum points of a function.

First-Derivative Test for Local Extrema Suppose c is a critical point for y = f(x) If f’ (x) > 0 throughout some interval (a, c) to the left of c and f’ (x) < 0 throughout some interval (c, b) to the right of c, then x = c is a local maximum point for the function f. AND

First-Derivative Test for Local Extrema Suppose c is a critical point for y = f(x) If f’ (x) 0 throughout some interval (c, b) to the right of c, then x = c is a local minimum point for the function f. AND

First-Derivative Test for Local Extrema Suppose c is a critical point for y = f(x) If f’ (x) > 0 (or f’ (x) < 0) throughout some interval (a, c) to the left of c and throughout some interval (c, b) to the right of c, then x = c is not a local minimum point for the function f.

Second Derivative Test Let f be a twice differentiable function in an interval I, and let c be an interior point of I. Then if f’ (c) = 0 and f’’ (c) < 0, then x = c is a strict local maximum point. if f’ (c) = 0 and f’’ (c) > 0, then x = c is a strict local minimum point. if f’ (c) = 0 and f’’ (c) = 0, then no conclusion can be drawn.

Example 1 Find the locations and values of all local extrema for the function with the graph

Example 2 Find the locations and values of all local extrema for the function with the graph

Example 3 Suppose that the graph to the right is the graph of f’ (x), the derivative of f(x). Find the locations of all relative extrema and tell whether each extremum is a relative maximum or minimum

Example 4 Find the critical points for the function below and determine if they are relative maximum or minimum points or neither.

Example 5 Find the critical points for the function below and determine if they are absolute maximum or minimum points or neither.

Example 6 Find the critical points for the function below and determine if they are absolute maximum or minimum points or neither.

Example 7 For the cost function and the price function find a.the number, q, of units that produces a maximum profit. b.the price, p, per unit that produces maximum profit. c.the maximum profit, P.

Example 8 Suppose that the cost function for a product is given by find the production level (i.e. value of x) that will produce the minimum average cost per unit.

In Summary To find local extrema, we need to look at the following types of points: i.Interior point in an interval I where f’ (x) = 0 ii.End points of I (if included in I) iii.Interior points in I where f’ (x) does not exist