MATH 1231 Chapter Notes - Chapter 3: Inflection Point, Maxima And Minima

If the derivative f z) is negative, increasing, concave up Then the function f(x) is (not all info may be used): O increasing, concave up increasing, concave down O decreasing, concave down decreasing, concave up Suppose F(t) has the derivative f(t) shown below, and F(0) 1. Find values for F(3) and F (8) Let A(z) f(t)dt, with fou) shown in the graph below. At what x values does A(x) have a local max r- At what r values does AC) have a local min: x Restrict your answers to values 0

Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f " to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points f(z) =: 4z4-32z' + 89z2-95z + 29 (a) Intervals of increase are at (b) Intervals of decrease are at (c) fhas a local maximum at a- (d) f has a local minimum at z = (e) The graph of f is concave up on the interval | (f) The graph of f is concave down on the interval (g) f has inflection points at z

Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to produeing graphs (with a graphing device) that display the major features of the curve. Use these graphs to locate the maximum and minimum values. r(x 1) f(z) has local minima at z = Preview f(a) has local maxima at Preview

7. The following is the graph of the derivative function f(x) of a differentiable function f defined on the interval (0, 11) 10 8 9 10 11 -10 -15 Note that the following questions are about the function/ mot the function r a. On what intervals is fincreasing? Decreasing? b. At what values ofx does fhave a local maximum? A local minimum? c. On what intervals is f concave up? Concave down? d. At what values of x does fhave a point of inflection? e. Assuming that f(O)-0, sketch a graph ofj