08Bsketchingcurves

Stationary Points
 * Curve sketching **

Recall that a ** stationary point ** occurs where the tangent is parallel to the x-axis. (ie the gradient of the tangent is zero).

In terms of derivatives, a stationary point occurs where f '(x) = 0.

In the graph to the right, **A**, **B** and **C** are stationary points.

The point **A** is called a ** __local minimum point__ **. {Notice that **A** is not the absolute minimum point of the graph, but it is the minimum point in the area __**local**__ to A}

The gradient table for point A would look like this: The point **C** is called a ** __local maximum point__ **. Stationary points **A** and **C** are also referred to as ** turning points **.

The point **B** is called a ** __stationary point of inflection__ **. It has a positive gradient on either side of the stationary point. The __point of inflection__ is the point on the graph where the gradient changes from concave down to concave up. In other words, the gradient was decreasing as it approached B and after B it begins increasing.

There are also two __non-stationary__ points of inflection on this graph (not marked) where the concavity changes. One is between A and B, the other is between B and C.

To the left of the first non-stationary point of inflection, the graph is always concave up. To the right of the second non-stationary point of inflection, the graph is always concave down. Between A and C the concavity changes three times (at each of the three points of inflection).

A ** stationary point of inflection ** could also have a negative gradient on either side of the stationary point (not shown in the graph). Locating Stationary Points


 * Example 1 **

(i) Sketch the following graphs, showing intercepts with the axes and stationary points. (ii) State the domain over which the function is strictly decreasing. math \\ \textbf{(a) } y = 2x^3 - 5x^2 +7 \\ \\ \textbf{(b) } y = \big( x^2 - 1 \big)^2 math

__**Solution:**__

(i) Find the stationary points and determine their nature.
 * (a) **y = 2x 3 – 5x 2 +7

math \dfrac{dy}{dx}=6x^2-10x math math \dfrac{dy}{dx}=0 \quad \text{ when } 6x^2-10x=0 math

math \\ 2x(3x-5)=0 \\ \\ x=0 \quad \textit{or} \quad x=\dfrac{5}{3} math



When x = 0, y = 7

math \text{When } x=\dfrac{5}{3}\quad y=2 \left( \dfrac{5}{3} \right)^3-5 \left( \dfrac{5}{3} \right)^2+7 math math .\quad\quad\quad\quad\;y=\dfrac{64}{27} math

Therefore, (0, 7) is a local maximum turning point, and (5/3, 64/27) is a local minimum turning point.

(ii) Find axial intercepts:

When x = 0, y = 7, the y - intercept is (0, 7).

When y = 0, 2x 3 – 5x 2 + 7 = 0

To factorise: find possible zeroes: –1, 1, –2, 2, –7, 7. P(–1) = 2(–1) 3 – 5(–1) 2 + 7 = 0

Therefore, (x + 1) is a factor. quadratic factor has no factors since

math \\ \Delta=(-7)^2-4\times2\times7 \\ .\quad=49-56<0 math

x-intercepts occur where f(x) = 0: (x + 1)(2x 2 – 7x +7) = 0 x + 1 =0 x = –1 hence (–1, 0) is the only x - intercept.



(iii) State the domain for which the function is strictly decreasing.

The function is strictly decreasing when the gradient is negative (including the endpoints of a negative section where it is zero)

math \dfrac{dy}{dx} \leqslant 0 \text{ when } \Big\{ x:0 \leqslant x \leqslant \dfrac{5}{3} \Big\} math


 * (b) ** y = (x 2 – 1) 2

(i) Find the stationary points and determine their nature.

math \\ \dfrac{dy}{dx}=2 \big( x^2-1 \big) \times 2x \\ \\ .\quad=4x \big( x^2-1 \big) math

math \\ \dfrac{dy}{dx}=0 \quad \text{ when } \quad4x=0 \; \textit{ or } \; x^2-1=0 \\ \\ x = 0, \;\; x = -1, \;\; x = 1 math



Therefore, (–1, 0) is a local minimum turning point, (0, 1) is a local maximum turning point and (1, 0) is a local minimum turning point.

(ii) Find axial intercepts.

When x = 0, y = 1, therefore (0, 1) is the y - intercept.

When y = 0, x 2 – 1 = 0, so x = ±1

Therefore, (–1, 0) and (1, 0) are x - intercepts.



(ii) The function is strictly decreasing when the gradient is negative or zero.

math \dfrac{dy}{dx} \leqslant 0 \text{ when } \big\{ x:x \leqslant -1 \big\} \textit{ and when }\big\{ x:0 \leqslant x \leqslant 1 \big\} math {**but not both at the same time**: __strictly decreasing__ means all points to the right of a given point, (a, b), must have a y-value less than b. To describe part of the graph as strictly decreasing, it must be one-to-one.} .