f x How can we determine algebraically where f is concave up and where f is concave down?
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1 Concavity Concave up and concave down Definition For a function f that is differentiable on an interval I, the graph of f is a. If f is concave up on a, b, then the secant line passing through points x 1, f x 1 and x, f x for any x 1 and x in a, b are above the curve y f x between x 1, f x 1 and x, f x. b. If f is concave down on a, b, then the secant line passing through points x 1, f x 1 and x, f x for any x 1 and x in a, b are below the curve y f x between x 1, f x 1 and x, f x x x a concave up function on 3, 3 a concave down function on 3, 3 Example The graph of f is given below. Determine graphically the interval on which f is a. concave up; b. concave up and decreasing x a. f is concave up on 1.8, 3.3 b. f is concave up and decreasing on 1.8,.5-1 f x How can we determine algebraically where f is concave up and where f is concave down? Theorem Suppose that f is differentiable on an interval I. The graph of f is 1
2 a. concave up on I, iff is increasing on I ; and b. concave down on I,iff is decreasing on I. Or, suppose that f exists on I. The graph of f is a. concave up on I, iff x for all x in I; b. concave down on I,iff x for all x in I.. Inflection points: Definition Suppose that f is continuous on the interval a, b. Letcbein a, b. Then the point c, f c is called an inflection point of f if the graph of f changes concavity at the point c, f c. Note that the graph of f changes concavity at the point c, f c if f changes from increasing to decreasing or from decreasing to increasing at c, f c or f changes from positive to negative or from negative to positive at c, f c. Example Let the graph of f x be given below. Find a. the x coordinate of each inflection point of f; b. where the graph of f is concave up and is concave down x f x a. f x when x, x, x 1 and x. f does not change sign at x. So, the x coordinates of inflection points of f are x, x 1 and x. b. f for x, x 1, x and f for x, 1 x. So, the graph of f is concave up on,, 1, and is concave down on, 1,. Example Let the graph of f x be given below. Find a. the x coordinate of each inflection point of f; b. where the graph of f is concave up and is concave down.
3 x -1 f x f x when x.8, x.5, x. f f is increasing for x.8,.5 x ; f f is decreasing for.8 x.5, x 5. a. So, x.8, x.5, x are the x coordinates of inflection points of f. b. The graph of f is concave up on,.8.5, and is concave down on.8,.5, 5. Example Let f x x 3 9x x 1.Find a. all inflection points of f; b. where the graph of f is concave up and is concave down. Verify your answers by graphing both f and f. a. i. Compute f : f x 6x 18x, f x 1x 18 1 x 3 ii. Solve f x : 1 x 3 x 3. iii. Check signs of f over intervals:, 3, 3, f 1 1 f 1 3, interval, 3 sign of f x 3, Since f changes sign at the point where x 3, 3, 79 is an inflection point of f. b. The graph of f is concave up on 3, and is concave down on, 3 f.. Check the graph of 3
4 x f,... f, -.- f 3. Second Derivative Test: Theorem Suppose that f is continuos on the interval a, b and f c, for some c in a, b. a. If f c, then f c is a local maximum and b. if f c, then f c is a local minimum. Example The graph of f is given below. Suppose that we know f 1,f and f. Determine if f 1, f and f are local maximum, local minimum or neither x -1 f x Example Let f x x 5 x. Find
5 a. the intervals of increase and decrease; b. all local extrema; c. the intervals of concavity; d. all inflection points; and e. sketch the graph of f based on the information in a.-d. The domain of f :,,, Compute f and f : f x 1 5 x x 5, x f x 5 x 3 5 x 3 Find critical numbers of f : f x x 5, x 5; f x is not defined x Determine the sign change of f over, 5, 5,,, 5, 5, f f 1 5, 16 Determine the sign change of f : f x, interval, 5 5,, 5 5, f x f x changes sign as x 3 changes sign. interval,, f x State the results: a. f is increasing on, 5, 5, and is decreasing on 5,,, 5. b. By the first derivative test, f 5 is a local maximum and f 5 is a local minimum. c. f is concave up on, and is concave down on,. d. Though the graph of f changes concavity at x, x is not an inflection point of f since it is not in the domain of f, so there is no inflection point e. Sketch the graph of f based on the information in a.-d x Example Let f x sinx cos x. Find a. the intervals of increase and decrease; b. all local extrema; c. the intervals of concavity; d. all inflection points; and 5
6 e. sketch the graph of f based on the information in a.-d. The domain of f : D f, Compute f and f : f x cos x sinx, f x sinx cos x Find critical numbers of f : f x cos x sinx, sinx cos x, tan x 1, x n, n, 1,,... Determine the sign change of f over... 7, 3, 3,,, 5, 5, 9,... f 1 f 1 1 f 1 1 f 1 1 interval... 7, 3 3,, 5 5, 9... sign of f Determine the sign change of f x sinx cos x : f x, sinx cos x, tan x 1, x 3 n Determine the sign change of f over... 5,,, 3, 3, 7, 7, 11,... f 1 f 1 1 f 1 1 f 1 1 interval... 5,, 3 3, 7 7, sign of f State the results: a. f is increasing on... 3,, 5, 9,... b. f c is a local maximum for c... 7,, 9,... f c is a local minimum for c... 3, 5,... c. f is concave up on... 5,, 3, 7,... and is concave down on..., 3, 7, d. Inflection points of f are: n e. Sketch the graph of f based on the information in a.-d 6
7 x y f x, - - y f x,... y f x 7
8 Example Sketch a graph of a function with the given properties: f f x, for all x; f 1 f x for x, f x for x, f Example Sketch a graph of a function with the given properties: 1. f, f 1 1, f 1 1 f x, for x 1 and x 1, f x for 1 x and x 1; f x for x and x 8
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