Identify the x-coordinates of, and classify, the stationary points of F.

Question

Identify the x-coordinates of, and classify, the stationary points of F if dy/dx = (x+3)⁹(x-1)⁵(2-x)⁶. Classify means you have to tell me whether they're relative max or relative min.

Answer 1

Hi Amir! In order to answer your question, I would first clarify that stationary point is a critical point, such that dy/dx = 0 or does not exist. In this problem, there is no such x-coordinate that dy/dx does not exist (there is no undefined result here). Therefore, we will only focus on when dy/dx = 0.

Using the Zero Product Property (basically, if a * b = 0, then a = 0, b = 0, or a = b = 0), then

dy/dx = 0 when (x + 3) ^ 9 = 0, (x - 1) ^ 5 = 0, and (2 - x) ^ 6 = 0 ⇒ x = -3, x = 1, x = 2

To know whether each of these points is a relative min or max, we need to know what values does dy/dx change. The x-coordinate at a stationary point will be a relative minimum if dy/dx changes value from negative to positive, and relative maximum when dy/dx changes value from positive to negative.

At x = -3, dy/dx changes from positive to negative, so x = -3 will be a relative max.

At x = 1, dy/dx changes from negative to positive, so x = 1 will be relative min.

At x = 2, dy/dx remains positive to positive, so there is neither a min or max.

I hope this helped you! Let me know if you want more help!