Finding stationary points from the derivative
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 orx = 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)
- At
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.
flashcards
| Question | Answer |
|---|---|
| How do you find the stationary points of a function? | By differentiating the function. |
| If | It is increasing (positive gradient). |
| If | It is decreasing (negative gradient). |
| If | It is stationary (zero gradient). |