Finding stationary points
Finding stationary points.
You can find stationary points on a curve by differentiating the equation of the curve and finding the points at which the gradient function is equal to 0.
One way of determining a stationary point.
The nature of the stationary point can be found by considering the sign of the gradient on either side of the point.
A stationary point can be:
- A local maximum, where the gradient changes from positive to negative (+ to -)
- A local minimum, where the gradient changes from negative to positive (- to +)
- A stationary point of inflection, where the gradient has the same sign on both sides of the stationary point
An alternative method for determining the nature of stationary points.
If you differentiate the gradient function, the result is called a second derivative.
At a stationary point:
- If the second derivative is positive, the point is a local maximum
- If the second derivative is negative, the point is a local minimum
- If the second derivative is 0, the stationary point could be a local minimum, a local maximum or a stationary point of inflection.
– (you need to look at the gradient on either side to find the nature of the stationary point)
Example using the second method:
y = x3 - x2 - 4x -1
find the values of the first and second derivatives where x= -1
dy/dx = 3x2 - 2x - 4 = (3 x -1 x -1) - (2 x -1) - 4 = 1
d2y/dx2 = 6x - 2 = (6 x -1) - 2 = -8
Since the second derivative (d2y/dx2) < 0, the point where x= -1 is a local minimum.
