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

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Answer 1 · 2018-08-10T13:09:13

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

#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]