Final answer:
To find the x value where a vertical tangent line exists for the function f(x) = 3(x-1)^(1/3), we can find the x value where the derivative of the function is undefined or infinite. Applying the power rule, we find the derivative of f(x) as f'(x) = (1/3) * 3(x-1)^(-2/3). Setting the derivative equal to infinity, we solve for x and find x = 28.
Step-by-step explanation:
A vertical tangent line occurs when the derivative of a function is undefined or infinite at a certain point. To find the x value at which a vertical tangent line exists for the function f(x) = 3(x-1)^(1/3), we need to find where the derivative of the function is undefined or infinite.
To find the derivative of f(x), we can use the power rule: d/dx (x^n) = nx^(n-1). Applying this rule, the derivative of f(x) is f'(x) = (1/3) * 3(x-1)^(-2/3).
We can set the derivative equal to infinity and solve for x: (1/3) * 3(x-1)^(-2/3) = infinity. Simplifying, (x-1)^(-2/3) = 3. Taking the cube of both sides, we get (x-1) = 27.
Therefore, the x value at which a vertical tangent line exists for the function f(x) = 3(x-1)^(1/3) is x = 28.