Suppose that a continuous function fhas a derivative f\' whose graph is shown be

Question

Suppose that a continuous function fhas a derivative f' whose graph is shown below over the interval (2, 11). y f'(x) -? 1 2/? 4 5 6 7 9 011 -2 -4 (a) Find the interval(s) over which f is increasing. (Enter your answer using interval notation.) Find the interval(s) over which f is decreasing. (Enter your answer using interval notation.) (b) Find the x-value(s) where fhas a local maximum. (Enter your answers as a comma-separated list.) Find the x-value(s) where f has a local minimum. (Enter your answers as a comma-separated list.)

Explanation / Answer

(a) Here from the graph since f'(x)>0 at x=(3,8) and x=(10,11) thus f(x) is increasing in the intervals (3,8) and (10,11). Similarly as f'(x)<0 at x=(2,3) and x=(8,10) thus f(x) is decreasing in the intervals (2,3) and (8,10).

(b) From the graph as the graph of f'(x) is going from negative to positive at points x=3 and x=10 i.e. f(x) goes from decreasing to increasing thus x=3,10 are points of local minima. Similarly as the graph of f'(x) goes from positive to negative at point x=8 i.e. f(x) goes from increasing to decreasing thus x=8 is the point of local maxima.

(c) From the graph since f'(x) is increasing in the intervals x=(2,4),(5,7)and (9,11) thus f(x) is concave upward in the intervals x=(2,4), (5,7) and (9,11).Similarly since f'(x) is decreasing in the intervals x=(4,5) and (7,9) thus f(x) is concave downward in the intervals x=(4,5) and (7,9).

(d) From the graph since f'(x) changes its nature at x=4,5,7,9 i.e. the graph goes from increasing to decreasing at points x=4,7 and decreasing to increasing at points x=5,9 which implies concavity of f(x) changes at points x=4,5,7,9. Thus the points of inflection of f(x) are x=4,5,7,9.

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