Question

(-5,2) (-2,-1) Graph of g The continuous function g has domain -5 < x < 2. The graph of g, consisting of two line segments and a semicircle, is shown in the figure above. The graph of g has a horizontal tangent at x = -1. Let h be the function defined by h(x) = S-29(t)dt for -5 < < 2. (a) Find the x-coordinate of each critical point of h on the interval -5 < x < 2. (b) Classify each critical point from part (a) as the location of a relative minimum, a relative maximum, or neither for h. Justify each of your answers. (c) For - 5 < < 2, on what open intervals is h increasing and concave up? Give a reason for your answer.

Answer 1

h(x) = ( 8(t)dt, -55x52 (a) h'(x) = g(x) h'(x) = 0 = g(x)=0 = x= -3.-11 So the critical points of g(x) are x=-3,-1,1 (6) "(x) = g'(x) "(-3) =g'(-3)=-1<0=x=-3 is relative maximum of h(x) h"(-1)= g'(-1)=0.5'(-1)=g(-1)<0 and h'(-1* )=g(-1*) < 0 =x=-1 is neither relative maximum nor relative minimum of h(x) (c) '(x) = g(x), h"(x)=g'(x) h'(x) > 0= g(x)>0=x+(-5-3), (1,2) Interval of increase : (-5.-3). (1.2) "(x) >0= g'(x)>0=x*(-2,-1).(0.2) Concave up : (-2,-1) U(0.2)