Final answer:
To find the values of x where the tangent line to the function f(x) = x³ - 7x² - 7x + 1 is horizontal, we need to find the points where the derivative of the function is equal to zero. The values of x where the tangent line to the function is horizontal are approximately -0.276 and 3.610.
Step-by-step explanation:
To find the values of x where the tangent line to the function f(x) = x³ - 7x² - 7x + 1 is horizontal, we need to find the points where the derivative of the function is equal to zero. The derivative of f(x) can be found by taking the derivative of each term separately: f'(x) = 3x² - 14x - 7.
Setting f'(x) equal to zero and solving for x, we get the quadratic equation 3x² - 14x - 7 = 0. Using the quadratic formula, x = (-(-14) ± sqrt((-14)² - 4(3)(-7))) / (2(3)). After simplifying, we find that x ≈ -0.276 and x ≈ 3.610.
The student is asking for all the values of x where the tangent line to the function f(x)=x³−7x²−7x+1 is horizontal. A horizontal tangent line occurs when the slope of the function is zero. Therefore, we need to find the derivative of the function, f'(x), and then solve for x when f'(x) = 0.
To find the derivative, we calculate: f'(x) = 3x² - 14x - 7. Setting this equal to zero gives us the quadratic equation 3x² - 14x - 7 = 0. We can solve this equation for x by factoring or using the quadratic formula. Once we find values of x that solve this equation, those will be the points where the function has a horizontal tangent line.