Question

state the intervals in which the function is increasing.
y=x^3/6

Answer 1

One way to find where it is increasing is to take the derivative and solve for the critical points, then test values between them to determine if they are increasing or decreasing:\[\frac{dy}{dx} = \frac{x^{2}}{2}\]\[0 = \frac{x^{2}}{2}\]\[0 = x\]So the only place where the curve is horizontal is at the origin. So just test a positive number and a negative number:\[\frac{(1)^{2}}{2} = \frac{1}{2}\]\[\frac{(-1)^{2}}{2} = \frac{1}{2}\]
So it is increasing everywhere (except exactly at the origin, where it is horizontal)