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 (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 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 3Suppose 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.
31292)( 23 xxxxf
Example 5
Find the critical points for the function below and determine if they are absolute maximum or minimum points or neither.
32
376)( xxf
Example 6
Find the critical points for the function below and determine if they are absolute maximum or minimum points or neither.
3)( 8 xexxf
Example 7For the cost function
and the price function
find
qqC 1480)(
qp 258
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 8Suppose 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 .
78138002.0)( 3 xxxC
)(xC
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