Analysis of Functions

Transcription

1 MATH 16 Analysis of Functions We now give an outline of the basic facts of derivatives that are used to analyze of a the graph of a function f ( x). It is always a good idea to first graph the function on your calculator to get a good overall view of the function s behavior. The First Derivative Evaluate f (x). Wherever f (x) > 0, then the slope of the tangent line to the graph of f ( x) is itive, which means that f ( x) is easing. Wherever f (x) < 0, then the slope of the tangent line to the graph of f ( x) is ative, which means that f ( x) is easing. We demonstrate with three quick examples below. f (x) = 0 when 2 x = 0 at x = 2 When x < 2, f (x) > 0, 2 so f ( x) is easing. f ( x) = 5 + x x 2 f (x ) = 2x When x > 2, f (x) < 0, so f ( x) is easing. f ( x) is easing for x in the interval (, 2). f ( x) is easing for x in the interval (2, ). f (x ) = 1 2 < 0 for all x 0 x f ( x) = 1 x f (x) = 1 x 2 < 0 So f ( x) is easing for all x 0. f ( x) is easing for x in the intervals (, 0) (0, ). f ( x) = ln x for x > 0 f (x) = 1 f > 0, for x > 0 x f (x ) = 1 x > 0 for x > 0 So f ( x) is easing for all x > 0. ( x) is easing for x in the interval (0, ).

2 Critical Points A critical point can occur one of two ways. A critical point is any value x for which (i) f (x ) = 0 or (ii) f (x ) is undefined but f ( x) is defined. The critical points give the sible x-values for which the function f ( x) is changing from easing to easing or changing from easing to easing. But such a change doesn't necessarily have to occur at each critical point. f() f ( x) = (8x x 2 ) 1/ f (x ) = 8 2x (8 x x 2 ) 2/ f (x ) = 0 when 8 2x = 0 So x = is a critical point. f (x ) is undefined when 8x x 2 = x(8 x ) = 0. So x = 0 and x = 8 are critical points. If x <, then f (x ) > 0, thus f ( x) is easing on (, ). If x >, then f (x ) < 0, thus f ( x) is easing on (, ). At x =, f () = 0 which means that the tangent line to the graph of f ( x) has 0 slope and f ( x) has flattened out. Because f ( x) is changing from easing to easing at x =, the function value f () = 16 1/ is the maximum value in that region. Even though x = 0 and x = 8 are critical points, they do not cause a change in the easing or easing nature of f ( x). Instead they are causing vertical tangents and changes in avity that we discuss below. Relative Extrema A relative maximum occurs at point c if f (x ) changes from itive to ative at c which causes f ( x) to change from easing to easing at c. The function value f (c) is the actual relative maximum value. This value f (c) may actually be the absolute maximum value of f ( x) if it is in fact the largest value of the entire graph and not just the maximum in the region around c. A relative minimum occurs at point c if f (x ) changes from ative to itive at c which causes f ( x) to change from easing to easing at c. The function value f (c) is the actual relative minimum value. This value f (c) may actually be the absolute minimum value of f ( x) if it is in fact the smallest value of the entire graph and not just the minimum in the region around c. The approximate numerical values of the critical points and the relative extreme values also can be found with the built-in max and min commands after graphing on a calculator.

3 -1 f ( x) = x + x 2 + 9x 6-1 f (x ) = x x + 9 = ( x 2 2x ) f (x ) is defined for all x f (x ) = 0 when x 2 2x = ( x + 1)( x ) = 0 So x = 1 and x = are critical points. At x = 1, f (x ) changes from ative to itive, so f ( x) changes from easing to easing and f ( 1) = 11 is a relative minimum value. It is not the absolute minimum value of the function because lim f ( x) =. x At x =, f (x ) changes from itive to ative, so f ( x) changes from easing to easing and f () = 21 is a relative maximum value. It is not the absolute maximum value of the function because lim f ( x) = +. x For this function, f is easing on (, 1) (, ), and f is easing on ( 1, ). Concavity If the rate of growth of f ( x) diminishes as x eases, then f ( x) is said to be ave. If the rate of growth of f ( x) eases as x eases, then f ( x) is said to be ave. Increasing, Concave Down Decreasing, Concave Down Decreasing, Concave Up Increasing, Concave Up Diminishes becomes more. becomes less. Increases But f (x ) gives the rate of growth of f ( x). So f ( x) is ave when f (x ) is easing. And f ( x) is ave when f (x ) is easing. To determine where f (x ) is easing/easing, we need to look at its derivative, which is f (x ), which is the second derivative of f ( x).

4 The Second Derivative, Concavity, and Inflection Points We now evaluate f (x ) and find the critical points of f (x ) in one of two ways. A critical point of f (x ) is any value x for which (i) f (x ) = 0 or (ii) f (x ) is undefined but f (x ) is defined. These critical points of f (x ) are used to determine the avity of the original function f ( x). If f (x ) > 0, then f (x ) is easing and f ( x) is ave. If f (x ) < 0, then f (x ) is easing and f ( x) is ave. If f (x ) changes from itive to ative, or from ative to itive, at point d then f ( x) is changing avity at point d. Then the point on the graph ( d, f (d )) is called an inflection point. We illustrate by determining the avity and inflection points for the functions in all the preceding examples.. 2 f ( x) = 5 + x x 2 f (x ) = 2x f (x ) = 2 f (x ) < 0 for all x, so f (x ) is easing for all x, and f ( x) is ave on the entire line (, ).,, f ( x) = 1 x f (x ) = 1 x 2 f (x ) = 2 x f (x ) < 0 for x < 0, so f (x ) is easing for x < 0, and f ( x) is ave on (, 0). f (x ) > 0 for x > 0, so f (x ) is easing for x > 0, and f ( x) is ave on (0, ).

5 f ( x) = ln x, for x > 0 f (x ) = 1 x, for x > 0 f (x ) = 1 x 2, for x > 0 f (x ) < 0 for x > 0, so f (x ) is easing for x > 0, and f ( x) is ave on (0, ). f() f ( x) = (8x x 2 ) 1/ f (x ) = 8 2x 0 8 (8 x x 2 ) 2/ f (x ) = 2(x 2 8x + 6) 9(8x x 2 ) 5/ f (x ) = 0 when x 2 8x + 6 = 0, but this quadratic has no real solutions. f (x ) is undefined when 8x x 2 = x(8 x ) = 0. So x = 0 and x = 8 are critical points of f (x ). f (x ) < 0 for x < 0 and for x > 8. f (x ) > 0 for 0 < x < 8. f ( x) is ave on (, 0) (8, ). f ( x) is ave on (0, 8). The points (0, 0) and (8, 0) are inflection points where f ( x) changes avity -1 1 f ( x) = x + x 2 + 9x 6-1 f (x ) = x x + 9 = ( x 2 2x ) 1 f (x ) = 6 x + 6 f (x ) = 0 when x = 1 When x <1, f (x ) > 0 and f ( x) is ave. When x >1, f (x ) < 0 and f ( x) is ave. f ( x) is ave on (, 1), and f ( x) is ave on (1, ). The point (1, f (1)) = (1, 5) is an inflection point where f ( x) changes avity.

6 The Second Derivative Test Se c is critical point of f ( x) for which f (c ) = 0. Then we can use the second derivative to determine whether f (c) is a relative maximum or relative minimum. If f (c ) = 0 with f (c) < 0, then f (c) is a relative maximum. f (c ) = 0 rel max f (c) < 0. If f (c ) = 0 with f (c) > 0, then f (c) is a relative minimum. f (c) > 0. rel min f (c) = 0 With the preceding example of f ( x) = x + x 2 + 9x 6, we had critical points from f (x ) = 0 of x = 1 and x =. Also, f (x ) = 6 x + 6. Because f ( 1) = 12 > 0, then f ( 1) is a relative minimum value of f. Because f () = 12 < 0, then f () is a relative maximum value of f.