![Test for Increasing or Decreasing
Functions
Let f be continuous on [a,b] and differentiable on (a,b).
1. If ( ) 0 for all x in (a,b), then f is increasing on [a,b].
2. If ( ) 0 for all x in (a,b), then f is decreasing on [a,b].
3. If ( ) 0 for all x in (a,b), then f is constant on [a,b].
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Lecture 12(point of inflection and concavity)
- 1. Sections 3.3-3.4 First and Second Derivative Information
- 2. Test for Increasing or Decreasing Functions Let f be continuous on [a,b] and differentiable on (a,b). 1. If ( ) 0 for all x in (a,b), then f is increasing on [a,b]. 2. If ( ) 0 for all x in (a,b), then f is decreasing on [a,b]. 3. If ( ) 0 for all x in (a,b), then f is constant on [a,b]. f x f x f x ′ > ′ < ′ =
- 3. Increasing/Decreasing To determine whether the function is increasing or decreasing on an interval, evaluate points to the left and right of the critical points on an f’ numberline. f '(x) c + __ inc __dec f '(x) c +__ incdec
- 4. First Derivative Test Let c be a critical number of a function f that is continuous on an open interval containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows… 1. If ( ) changes from negative to positive at c, then f(c) is a relative minimum. 2. If ( ) changes from positive to negative at c, then f(c) is a relative maximum. f x f x ′ ′
- 5. 1st Derivative Test f '(x) c + inc __dec If the sign changes from + to - at c, then c is a relative maximum. Max f '(x) c +__ incdec If the sign changes from - to + at c, then c is a relative minimum. Min
- 6. • A curve is concave up if its slope is increasing, in which case the second derivative will be positive ( f "(x) > 0 ). • Also, the graph lies above its tangent lines. •A curve is concave down if its slope is decreasing, in which case the second derivative will be negative (f "(x) < 0 ). • Also, the graph lies below its tangent lines. Concavity
- 7. Test for Concavity Let f be a function whose 2nd derivative exists on an open interval I. 1. If ( ) 0 for all x in I, then f is concave upward. 2. If ( ) 0 for all x in I, then f is concave downward. f x f x ′′ > ′′ <
- 8. Concavity Test To determine whether a function is concave up or concave down on an interval, determine where f "(x) = 0 and f "(x) is undefined. Then evaluate values to the left and right of these points on an f " numberline. f "(x) c + __ ccu __ccd f "(x) c +__ ccuccd
- 9. A point where the graph of f changes concavity, from concave up to concave down or vice versa, is called a point of inflection. At a point of inflection the second derivative will either be undefined or 0. Inflection
- 10. If the signs on the f "(x) numberline do not change, then c is not an inflection point. f "(x) c __ ccdccd __ Not inflection When the signs change on an f " numberline, there is an inflection point.
- 11. Second Derivative Test Let f be a function such that f’(c) = 0 and the 2nd derivative of f exists on an open interval containing c. 1. If ( ) 0 , then f(c) is a relative minimum. 2. If ( ) 0,then f(c) is a relative maximum. 3. If ( ) 0, then the test fails. Use the 1st Derivative Test. f c f c f c ′′ > ′′ < ′′ =