Where are the stationary points?

Where are the stationary points?

The stationary points of a function are where the (first) derivative is equal to zero.

Example: find the stationary points of y=x^3-3x^2+4

• \frac{dy}{dx} = 3x^2 - 6x
• Set \frac{dy}{dx} = 0:
  • 3x^2 - 6x = 0
  • 3x(x - 2) = 0
  • x = 0 or x = 2
• Substitute back to find y-coordinates:
  • At x = 0: y = 0^3 - 3(0)^2 + 4 = 4 → stationary point at (0, 4)
  • At x = 2: y = 2^3 - 3(2)^2 + 4 = 8 - 12 + 4 = 0 → stationary point at (2, 0) Answer: (0, 4) and (2, 0)

Derivative

We find the first derivative by differentiating the function.

What does the derivative mean?

• If \frac{dy}{dx} > 0, the function is increasing (as x increases, y increases)
• If \frac{dy}{dx} < 0, the function is decreasing (as x increases, y decreases)
• If \frac{dy}{dx} = 0, the function is stationary

Finding the type of stationary point

See classifying stationary points.