Does the graph of the function y\ =\ 3\sqrt[3]{x-1}show that the function is increasing or decreasing?
To determine if the function is increasing or decreasing, we need to examine the slope of the graph at different points.
The derivative of the function can tell us the slope of the graph. Taking the derivative of y = 3√[3]{x - 1}, we get:
dy/dx = (√[3]{x - 1})' * 3 = (1/3(x - 1)^(-2/3)) * 3 = (x - 1)^(-2/3)
The derivative is positive when (x - 1)^(-2/3) > 0 or (x - 1)^2 > 0.
Since (x - 1) is the base and it is always positive for this graph, (x - 1)^(-2/3) will also always be positive. Therefore, the derivative is always positive, indicating that the function is increasing.