For what values of x is #f(x)=(2x-2)(x-3)(x+3)# concave or convex?

1 Answer
Aug 10, 2018

#x in (-oo, 1/3); f(x)# is concave and
#x in (1/3,oo); f(x)# is convex.

Explanation:

#f(x)=(2x-2)(x-3)(x+3) # or

#f(x)= (2x-2)(x^2-9)#

#f^'(x)= 2(x^2-9)+ (2x-2)*2x# or

#f^'(x)= 2x^2+4 x^2-4x-18# or

#f^'(x)= 6 x^2 -4 x -18 #

#f^''(x)= 12 x -4 ; f^''(x)=0 or 12 x- 4 =0 or x = 1/3 #

Let’s select a convenient number in the interval less and

more than #1/3 ; x= 0 and 1:. f^"(0)= -4 ; (<0):. #

(concave down) and #f^''(1)=8 ; (>0):.# concave up.

Therefore, #x in (-oo, 1/3); f(x)# is concave and

#x in (1/3,oo); f(x)# is convex.

graph{(2x-2)(x-3)(x+3) [-80, 80, -40, 40]} [Ans]