3.5 - Concavity 1. Concave up and concave down
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1 . - Concavit. Concave up and concave down Eample: The graph of f is given below. Determine graphicall the interval on which f is For a function f that is differentiable on an interval I, the graph of f is If f is concave up on a, b, then the secant line passing through points, f and, f for an and in a, b are above the curve f between, f and, f. If f is concave down on a, b, then the secant line passing through points, f and, f for an and in a, b are below the curve f between, f and, f ()concave up; () concave up and decreasing. () f is concave up on.8, f.7 -. () f is concave up and decreasing on.8, a concave up. - a concave down How can we determine algebraicall where f is concave up and where f is concave down? Theorem: Suppose that f is differentiable on an interval a, b. Then the graph of f is (a) concave up on a,b, if f is increasing on a, b ; and (b) concave down on a,b, iff is decreasing on a, b. Or, suppose that f eists on a,b. The graph of f is (a) concave up on a,b, if f for all in a, b ; (b) concave down on a,b, iff for all in a, b.. Inflection points: Definition: Suppose that f is continuous on the interval a, b. Let c be in a, b. Then the point c, f c is called an inflection point of f if the graph of f changes concavit at the point c, f c. Note that: changes concavit at c, f c f changes from increasing to decreasing at c, f c f changes from positive to negativeor from negative to positive at c, f c. Eample: Let the graph of f be given at right. Find () the coordinate of each inflection point of f; () where the graph of f is concave up f () f when,, and. f does not change sign at. So, the coordinates of inflection points of f are, and. () f for,, and f for,. So, the graph of f is concave up on,,,...
2 Eample: Let the graph of f be given at right. Find () the coordinate of each inflection point of f; () where the graph of f is concave up. - - f f when. 8,.,. f f is increasing for.8,. ; f f is decreasing for.8.,. () So,. 8,., are the coordinates of inflection points. () The graph of f is concave up on,.8.,. Eample: Let f 9. Find () all inflection points of f; () where the graph of f is concave up and is concave down. Verif our answers b graphing both f and f. (i.) Compute f : f 6 8, f 8 (ii.) Solve f :. (iii.) Check signs of f over intervals:,,, f interval,,, f sign of f Since f changes sign at the point where,, 79 is an inflection point of f. The graph of f is concave up on, and is concave down on,. Verif the results with the graph of f. 6. Second Derivative Test: Theroem: 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 maimum and (b) if f c, then f c is a local minimum red f,green f,blue f. Eample: The graph of f is given below. Suppose that we know f, f and f. Determine if f, f and f are local maimum, local minimum or neither. f and f,...7 so f and f are local maimum values. - f, f is a local minimum value. - f 7 8
3 Eample: Let f /. Find () the intervals of increase and decrease; () all local etrema; () the intervals of concavit; () all inflection points; and () sketch the graph of f based on the information in a.-d. The domain of f :, Compute f and f : f / Find critical numbers of f : tpe (i): f f 9 / 9 9,,,.7,. tpe (ii): f is not defined :. Determine the sign change of f over,. 7,.7,,,.,., f, f.867 f... 9, f interval,.7.7,,.., f Find where f : 9.. Find where f is not defined: Determine the sign change of f over intervals:, and, f 9 f 9 interval,, f () Sketch the graph of f based on the information in a.-d State the results: () f is increasing on,.7,., and is decreasing on.7,,,.. () B the first derivative test, f is a local maimum and f is a local minimum. () f is concave up on, and is concave down on,. () f changes concavit at and is in the domain of f so it is an inflection point of f.
4 Eample: Let f e cos. Find () the intervals of increase and decrease; () all local etrema; () the intervals of concavit; () all inflection points; and () sketch the graph of f based on the information in a.-d. The domain of f : D f, Compute f and f : f e cos e sin e cos sin f e cos sin e sin cos e sin Find critical numbers of f : tpe (i): f e cos sin, cos sin, sin cos, tan n, n,,,... tpe (ii): f is not defined: None. Determine the sign change of f over... 9,,,,,,, 7, 7, f e cos e, f e cos e f, f e cos e, f e cos e interval 9,,,, sign of f Find where f :e sin, n, n,,,... Find where f is not defined: None. Determine the sign change of f e sin over intervals:...,,,,,,, f e / sin, f e / sin f e / sin, f e / sin. 7. interval,,,,. sign of f -. State the results: () f is increasing on... 9,,,,, () f c is a local maimum for c,, f c is a local minimum for c..., () f is concave up on...,,, and is concave down on...,,,... () Inflection points of f are: n () Sketch the graph of f based on the information in ()-() 6
5 Eample: Sketch a graph of a function with the given properties: (i) f (ii) f, for all ; f (iii) f for, f for, f Eample: Sketch a graph of a function with the given properties: (i) f, f, f (ii) f, for and, f for and ; (iii) f for and 7
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