The graph of a function f is given. Consider the function below when answering t

Question

The graph of a function f is given. Consider the function below when answering the following questions. g (x) = integral_0^x f (t)dt. At what values of x do the local maximum and minimum values of g occur? (Enter solutions from smallest to largest If there are any unused answer boxes, enter NONE in the last boxes.) x_1min = 0 x_2min = 16 x_1max = NONE x_2 max = NONE Where does g attain its absolute maximum value? x=_______ On what intervals is g concave downward? (Enter the intervals that contain smaller numbers first.) (_________, ____________) Union (_________, ____________) Union (_________, ____________)

Explanation / Answer

g'(t) = f by theorem of Calculus

g has max or minima when f =0

Hence at points 0, s/2, 3s/2, 5s/2, 7s/2, 9s/2

f' is negative x = s/2/ 5s/2 and 9s/2

x1min = 0

x2min = 3s/2

x3min=7s/2

x1max = s/2

x2max = 5s/2

x3max = 9s/2

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c) g is concave up when f"(x) >0 i.e. when the given graph is increasing

i.e. at (0,s/4) U (s,2s) U (3s,4s)

Concave down in (s/4,s) U (2s,3s) U(4s, 9s/2)

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