09Fstationarypoints

= Stationary Points =


 * A ** stationary point ** is the place on a graph where the tangent to the curve is horizontal
 * ie the gradient equals zero
 * ie the derivative 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 1 **

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

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


 * ** 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.


 * They are called Local because there may be other points in the graph which are higher or lower

**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)
 * the curve is concave down
 * 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.
 * the curve is concave up


 * 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 2 **

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


 * the stationary point occurs at .. (2, –3)


 * Now construct a gradient table for values close to x = 2
 * ** Ensure the first row is in __increasing__ order **
 * ** The second row shows the derivative for that x-value **
 * ** eg at x = 1, derivative = –2 **
 * **The third row is a short line segment**
 * **with a slope either negative, zero or positve**
 * **eg at x = 1, derivative is negative, so line slopes down**

... ..
 * 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)


 * Note: **
 * In a gradient table, the steepness of the slope doesn't matter
 * So sometimes in the second row, we only write –, 0 or + instead of the actual value


 * Locating Turning Points with your Calculator **


 * 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}


 * In the example shown on the right
 * a local maximum has been found for the curve .. y = sin(x)
 * the local maximum occurs at (1.57, 1) .. (approx)

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