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Stationary Points (Maximum and Minimums) and Differentiation

On a graph a stationary point is any point where the gradient is 0 so where the graph is flat. For example the graph y=x2 has one stationary point at the origin.

Finding the Stationary Points

We know that stationary point occur when the gradient is 0 so when the derivative of the graph is 0, so in order to find the stationary points we but first differentiate the curve.

For example lets consider the graph $y = 3x^2 + 2x - 7$. We cab differentiate this to find
$\frac{dy}{dx} = 6x + 2$

We must then equate the derivative to 0 and solve the resulting equation. This is because we are trying to find the points where the gradient is zero and these point occur exactly at the solutions of the equation we have formed.

So in our example we form the equation
$6x + 2 = 0$
by equating our expression for $\frac{dy}{dx}$, $6x + 2$, to 0
Solving this equation we find that stationary points occur exactly when
$x = \frac{2}{6} = \frac{1}{3}$
Note that there can be more than solution to this equation, each of which is a valid stationary point.

Finally we should also find the y co-ordinate for the stationary point by putting this value of x into the initial equation. So for this example $y= 3 \cdot \frac{1}{3}^2 + 2 \cdot \frac{1}{3} - 7 = -6$
So the only stationary point is at $(\frac{1}{3},-6)$

Nature of Stationary Points

The nature of a stationary point simply means what the graph is doing around it and are characterised by the second derivative, $\frac{d^{2}y}{ dx^2}$ (found by differentiating the derivative). There are three types of stationary point:

1. Maximum Points: These are stationary points where the graph is sloping down on either side of the stationary point (a sad face type of curve).
Here ${d^{2}y}{dx^2} < 0$
2. Minimum Points: These are stationary where the graph is sloping upwards on either side of the point (a happy face)

Here ${d^{2}y}{dx^2} > 0$
3. Point of Inflection: Here the direction of the slope of the graph is the same either side of the stationary point, it can be in either direction.

At a point of inflection ${d^{2}y}{dx^2} = 0$ but ${d^{2}y}{dx^2} = 0$ isn’t enough to ensure that a point really is a point of inflection as it could still be a maximum or minimum point
4. Checking the nature of a Stationary Point when ${d^{2}y}{dx^2} = 0$
In this case the easiest thing to do is look a small distance either side of the point and see whether the y value is greater than or less than that of the stationary point. You can then draw yourself a picture to see what it is. For example if they are both greater than the stationary point you know it is a minimum point, but if one is greater and one is less than it is a point of inflection

Warning: checking points either side does not guarantee the correct result as there may be another stationary point or a break in the graph between where you are checking and the stationary point so you should always check using the derivatives if possible

Categories: algebra, calculus