Determine the value(s) of x for which the function f (x) = x3 – 12x + 3 has horizontal tangentlines. Show your work.

To find the x-values where the function f(x) = x^3 - 12x + 3 has horizontal tangent lines, we need to find where the derivative of the function is equal to zero.

Step 1: Find the derivative of the function. The derivative of f(x) = x^3 - 12x + 3 is f'(x) = 3x^2 - 12.

Step 2: Set the derivative equal to zero and solve for x. 0 = 3x^2 - 12 Divide both sides by 3 to simplify: 0 = x^2 - 4 This is a difference of squares, so it can be factored: 0 = (x - 2)(x + 2)

Step 3: Solve for x. Setting each factor equal to zero gives the solutions x = 2 and x = -2.

So, the function f(x) = x^3 - 12x + 3 has horizontal tangent lines at x = 2 and x = -2.