How do I find the co-ordinates and nature of the stationary points on a curve?
The stationary points on a curve are the exact point of a maximum, minimum or point of inflection. They can be defined as points where the gradient, dy/dx = 0. You can show this by drawing a tangent to the curve at such points, this tangent will just be a horizontal line, where the change in “y” is 0. So if you have a question which asks you to find the co-ordinates of the stationary points of a curve, your first step is to differentiate the curve’s equation, find dy/dx and equate it to zero. The vast majority of questions that you encounter at this level will now give you a quadratic equation, so step two is to solve that equation and find two x values. These are the x-coordinates of the stationary points. To find the corresponding y values, you need to substitute the x values found into the equation of the curve in question.To determine the nature of these stationary points, that is, whether they are a maximum, minimum or a point of inflection, you must find [d2y/dx2]. You do this by differentiating the dy/dx found earlier on. You then substitute the x values we found for the stationary points into the [d2y/dx2] function, if [d2y/dx2] < 0 for a stationary point, then that point is a maximum, if [d2y/dx2] > 0 then the point is a minimum. The explanation behind this is that [d2y/dx2] is the rate of change of the gradient, if you look at the peak of a curve when drawn on axis, you can see that the gradient before maximum is positive and the gradient afterwards is negative, therefore the rate of change of the gradient has to be negative and [d2y/dx2] < 0. The opposite is true for a minimum. If [d2y\dx2] = 0, then the point could be a maximum, minimum or a point of inflection, further investigation is required.
