Definition of Increasing and Decreasing A function f(x) is increasing on an interval if for any two numbers x 1 and x in the interval with x 1 < x, then f(x 1 ) < f(x ). As x gets larger, y = f(x) gets larger. A function f(x) is decreasing on an interval if for any two numbers x 1 and x in the interval with x 1 < x, then f(x 1 ) > f(x ). As x gets larger, y = f(x) gets smaller. Test for Increasing and Decreasing Theorem 5 Let f(x) be continuous on [a, b] and differentiable on (a, b). 1. If f (x) > 0 for all x in (a, b), then f(x) is increasing on [a, b].. If f (x) < 0 for all x in (a, b), then f(x) is decreasing on [a, b]. 3. If f (x) = 0 for all x in (a, b), then f(x) is constant on [a, b]. Proof Use the following for each case. Pick two x values in [a, b]. Apply the Mean Value Theorem. Use the definition of continuity for the end points. 1. If f (x) > 0 for all x in (a, b), then f(x) is increasing on [a, b].. Similar argument 3. Similar argument Finding Intervals of Increasing and Decreasing To find the intervals where f(x) is increasing or decreasing, 1. Find the critical numbers.. Set up test intervals using the critical numbers and undefined points. 3. Determine the sign of f (x) in each interval. 4. Use Theorem 5 to determine whether f(x) is increasing or decreasing. (or possibly constant)
First Derivative Test Theorem 6 Let c be a critical number of a function f(x) that is continuous on the open interval I containing c. If f(x) is differentiable on the interval I, except possibly at c, then f(c) can be classified as follows 1. If f (x) changes from negative to positive at c, then f(c) is a relative minimum.. If f (x) changes from positive to negative at c, then f(c) is a relative maximum. 3. If f (x) doesn t change sign at c, then f(c) is neither a relative maximum nor minimum. Proof Use Theorem 5. Examples (from 3.1) a. Where are the following functions increasing and decreasing? b. What are the relative extrema for the following functions? f(x) = x 3 x 4x + 5 x x 3 4 4 crit num: x = -/3 or x 3x 1 x 4 x 4 x 8x 11 ( x 4) crit num: x = 4 3 3
f(x) = sinx on [0, π] f(x) = x 3 if x > 3/ cos x if x < 3/ (n+1) dne if x = 3/ crit num: x = crit num: x = 3/
3 5x 7 x x 3 5 3(5x 7) /3 x 3 or x -1 x 1 ( x x 3) 1/ crit num: x = -7/5 crit num: x = -1 or 3 Definition of Concave Up and Concave Down Let f(x) be differentiable on (a, b). A function f(x) is concave up on (a, b) if f (x) is increasing on (a, b). How do you determine if f (x) is increasing? A function f(x) is concave down on (a, b) if f (x) is decreasing on (a, b). How do you determine if f (x) is decreasing?
Test for Concave Up and Concave Down Theorem 7 Let f(x) be twice differentiable on (a, b). 1. If f (x) > 0 for all x in (a, b), then f(x) is concave up on (a, b).. If f (x) < 0 for all x in (a, b), then f(x) is concave down on (a, b). Proof Follows directly from Theorem 5 and the definition of concavity. Finding Intervals of Concave Up and Concave Down To find the intervals where f(x) is concave up and concave down, 1. Find critical numbers of f (x). f (x) = 0 and f (x) does not exist. Set up test intervals using the critical numbers and undefined points. 3. Determine the sign of f (x) in each interval. 4. Use Theorem 7 to determine f(x) is concave up or concave down. Point of Inflection The point (c, f(c)) is a point of inflection if the tangent line exists (including a vertical tangent line) and the graph changes concavity at the point. Application of Critical Numbers Theorem 8 If f(x) has a point of inflection at x = c, then f (x) = 0 or f (x) does not exist. Proof Similar to Theorem.
Examples a. Where are the following functions concave up and concave down? b. What are the inflection points of the following functions? f(x) = x 3 x 4x + 5 3x 4x 4 x 3x 1 x 4 x 4 x 8x 11 ( x 4)
f(x) = sinx on [0, π] f(x) = x 3 if x > 3/ cos x dne if x < 3/ if x = 3/
3 5x 7 x x 3 domai n: 5 3( 5 x 7) /3 x 3 or x -1 x 1 ( x x 3) 1/
Second Derivative Test Theorem 9 Let f (c) = 0 and the second derivative of f(x) exists on an open interval containing c. 1. If f (c) > 0, then f(c) is a relative minimum.. If f (c) < 0, then f(c) is a relative maximum. 3. If f (c) = 0, use first derivative test. Proof Apply the alternate definition to the second derivative and use the first derivative test. Examples Use the second derivative test, if it applies, to find the relative extrema. a. f(x) = x 3 3x + 3 b. x 4 x
c. f(x) = cosx x on [0, 4π]