Find and graph all intercepts, vertical asymptotes, critical points, and inflection points of the function. Label each point with its exact coordinates.
f (x) = ( x - 2 )3
This function is 0 when x = 2, so the x-intercept is (2,0). The y-intercept is (0,-8).
The derivative is
f'(x) = 3( x - 2 )2
which is also zero when x = 2, so the x-intercept (2,0) is also a critical point.
The second derivative is
f"(x) = 6( x - 2 ) which is also zero when x = 2. Since f" changes sign at x = 2, (2,0) is also an inflection point.