1YStationaryPoints

= Stationary Points =


 * A ** stationary point ** is the place on a graph where the tangent to the curve is horizontal
 * ie the gradient equals zero.


 * The stationary points for any function can be found by making the derivative equal to zero.

math . \qquad \quad \text{Therefore, we want } \;\; \dfrac{dy}{dx} = 0 math

math . \qquad \text{Recall that for any polynomial } \;\; y = ax^n math

math . \qquad \quad \text{The derivative is } \;\; \dfrac {dy}{dx} = nax^{n-1} math

Example

math . \qquad \text{Find the derivative of } \;\; y = 4x^3 - 12x \\.\\ . \qquad \text{and hence find the location of any stationary points} math
 * Solution: **

math . \qquad \dfrac {dy}{dx} = 12x^2 - 12 math


 * ** Hence for this function, the stationary points occur at: **

math \\ . \qquad \dfrac{dy}{dx} = 0 \\. \\ . \qquad \Longrightarrow \; 12x^2 - 12 = 0 \\. \\ . \qquad \qquad x^2 - 1 = 0 \\.\\ . \qquad \qquad \big( x + 1 \big) \big( x - 1 \big) = 0 \\.\\ . \qquad \qquad \quad x = -1 \quad or \quad x = 1 \\.\\ math

math \\ . \qquad \text{at } x = -1, \quad \Rightarrow \quad y = 8 \\.\\ . \qquad \text{at } x = 1, \; \quad \Rightarrow \quad y = -8 \\.\\ . \qquad \text{Stationary points occur at } \; \big( -1, \; 8 \big) \quad and \quad \big( 1, \; -8 \big) math


 * Types of Stationary Point **:

There are 3 types of stationary point
 * 1) Local Minimum
 * 2) Local Maximum
 * 3) Stationary Point of Inflection




 * Turning Points **


 * Local Minimums and Local Maximums are also called ** turning points **.
 * The gradient "turns" from positive to negative (or negative to positive) on each side of the turning point.

**Stationary Points of Inflection**


 * The gradient has the __ same __ sign on each side of a ** point of inflection **.
 * either positive and positive (as shown above) or negative and negative (not shown)


 * A __**point of inflection**__ is the point where the curve of the graph changes from concave down to concave up (or vice versa).


 * The example in the table above shows that the gradient is positive but decreasing as we approach the stationary point (from the left)
 * Then at the stationary point the gradient reaches zero.
 * Then as we move to the right of the stationary point, the gradient begins increasing again.


 * We will also be studying __**non-stationary points of inflection.**__
 * These are points where the concavity changes without the gradient ever becoming zero.


 * Gradient Table ** :


 * Once you have located the x-coordinate of a stationary point, the nature of the stationary point can be determined by constructing a gradient table.
 * This involves finding the derivative (gradient) a small step on each side of the stationary point.


 * Example **

math . \qquad \text{Find the coordinates and nature of the stationary points } \;\; y = x^2 - 4x + 1 math


 * Solution**


 * Find the derivative

math . \qquad \dfrac {dy}{dx} = 2x - 4 math


 * Make the derivative equal zero to find the stationary point

math \\ . \qquad \dfrac{dy}{dx} = 0 \\. \\ . \qquad 2x - 4 = 0 \\ .\\ . \qquad \quad \; 2x = 4 \\ .\\ . \qquad \quad \;\; x = 2 \\ .\\ math


 * y-coordinate of stationary point
 * sub x = 2 into the rule

math \\ . \qquad y = (2)^2 - 4(2) + 1 \\ .\\ . \qquad y = 4 - 8 + 1 \\ .\\ . \qquad y = -3 math


 * Now construct a gradient table for values close to x = 2
 * ** Ensure the first row is in increasing order **

... ...


 * Inspection of the third row of the table reveals that the stationary point is a __local minimum__.


 * Coordinates of the local minimum are : (2, –3)


 * Locating Turning Points with your Calculator **


 * The Specialist Maths text gives a more complicated way of doing this in Eg 4


 * 1) Enter and sketch the function in the Graph&Table section of your calculator
 * Enter the equation at Y1= then click on the graph icon (1st on left)
 * 1) Make sure the window is sized so that the turning points are visible
 * Use the ZOOM menu, or the resize icon (3rd from the left)
 * 1) Use the built-in functions to find the coordinates of the turning point
 * {**Min** and **Max** are in the ANALYSIS menu, G-SOLVE submenu}


 * Example **

Finding the maximum point in y = sin(x) is shown to the right.
 * The local maximum is found to be (1.571, 1)
 * Notice that this is the decimal approximation.
 * To get the exact value, you would need to use a different method.

Finding the Exact Value of a Stationary point on the calculator


 * In the main section of the calculator, solve the equation formed by setting the derivative to 0.

math . \qquad \textbf{solve} \left( \dfrac{d}{dx} \big( \sin(x) \big) = 0, \; x \right) math
 * This can be done by entering:

math .\qquad x = \dfrac{\pi}{2} math
 * [[image:1Ycalc2.gif width="290" align="right"]]
 * In the example, the first solution is where the constant {constn(1) } = 0, so

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