Find the values of x for which the following are satisfied. (Enter your answers as comma-separated lists.)

Question

Let f(x) = 2/3x³+x²-12x+1

(a) f’(x) = -12

X=?

(b) f’(x) = 0

X=?

(c) f’(x) = 12

X=?

Answer 1

f(x) = (2/3)x^3 + x^2 - 12x + 1

f'(x) = 2x^2 +2x -12

f'(x) = -12 when x= 0

or when 2x^2+2x = 0

factor, divide by 2

x(x+1) = 0

x= 0 and x+1 = 0 or x=-1

x = -1, 0

f'(x) = 0 =2x^2 -2x -12

divide by 2

x^2 -2x -6 = 0, use the quadratic formula

x = 2/2 + or - (1/2)sqr(4+24) = -1 + or -sqr7

x = -1-sqr7, -1+sqr7

f'(x) = 2x^2 -2x -12 = 12

2x^2 -2x -24 = 0

x^2 -x - 12 = 0 factor and set each factor = 0

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

x-4=0 or x+3 = 0

x =-3, 4

Answer 2

I will do A, leaving B and C for you

Let f(x) = 2/3x³+x²-12x+1

f'(x)=2x^2+2x-12

2x^2+2x-12=-12

2x^2+2x-12+12=0

2x^2+2x=0

x^2+x=0

x(x+1)=0

x=0,-1