Suppose the function g(x) has a domain of all real numbers. The second derivativ

Question

Suppose the function g(x) has a domain of all real numbers. The second derivative of g(x) is shown below. g"(x) = (x - 3)5 (x + 4)(x - 2)4 Give the intervals where g(x) is concave down. (Enter your answer using interval notation. If an answer does not exist, enter DNE.) Give the intervals where g(x) is concave up. (Enter your answer using interval notation. If an answer does not exist, enter DNE.) Find the x-coordinates of the inflection points for g(x). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) x =

Explanation / Answer

a)g(x) concave down when g"(x)<0

==>(x-3)5(x+4)(x-2)4<0

==>(x+4)(x-3)5(x-2)4<0

==>x+4>0 or x-3<0

==>x>-4 ,x<3

==>x=(-4,3)

b)g(x) concave up when g"(x)>0

==>(x-3)5(x+4)(x-2)4>0

==>(x+4)(x-3)5(x-2)4>0

==>x+4<0 or x-3>0

==>x<-4 ,x>3

==>x=(-infinity,-4)U(3,infinity)

c)inflection point ==>g"(x)=0

(x-3)5(x+4)(x-2)4=0

==>x=3,-4,2

but g"(x) doesnot change its sign in the neighbourhood of x=2. i.e,g"(1.999)<0 and g"(2.001)<0. so no inflection point at x=2

inflection point only at x=-4,x=3

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