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

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.

To determine the concavity of the function \( f(x) = (2x - 2)(x - 3)(x + 3) \), you need to find its second derivative and then analyze its sign: 1. Find the first derivative of \( f(x) \): \[ f'(x) = 2(2x - 2)(x - 3) + (x - 3)(x + 3) + (2x - 2)(x + 3) \] 2. Simplify the first derivative: \[ f'(x) = 2(x - 3)[2(x - 2) + (x + 3) + 2(x + 3)] \] \[ f'(x) = 2(x - 3)[2x - 4 + x + 3 + 2x + 6] \] \[ f'(x) = 2(x - 3)(5x + 5) \] \[ f'(x) = 10(x - 3)(x + 1) \] 3. Find the second derivative of \( f(x) \): \[ f''(x) = 10[(x - 3)(1) + (x + 1)(1)] + 10(x - 3)(1) \] \[ f''(x) = 10[(x - 3) + (x + 1) + (x - 3)] \] \[ f''(x) = 10(3x - 5) \] \[ f''(x) = 30x - 50 \] 4. Analyze the sign of \( f''(x) \) to determine concavity: - \( f''(x) > 0 \) implies the function is concave up. - \( f''(x) < 0 \) implies the function is concave down. 5. Solve for \( x \) when \( f''(x) = 0 \) to identify possible points of inflection: \[ 30x - 50 = 0 \] \[ 30x = 50 \] \[ x = \frac{50}{30} \] \[ x = \frac{5}{3} \] 6. Therefore, \( f(x) \) is concave up when \( x < \frac{5}{3} \) and concave down when \( x > \frac{5}{3} \).