Back to: Further Mathematics SS2

**Rate of Change**

If y = *f*(x), can sometimes be interpreted as the rate at which y is changing with respect to x. if *y* increases as x increases, 0, while if *y* decreases as x increases, 0.

**Example**

The radius of a circle is increases a t the rate of 0.01cm/s. find the rate at which the area is increasing when the radius of the circle is 5cm.

**Example**

Water is leaking from a hemisphere bowl of radius 20cm at the rate of 0.5cm^{3}/s. Find the rate at which the surface area of the water is decreasing when the water level is half-way from the top.

**MAXIMUM AND MINIMUM POINTS **The points on a curve at which = 0, are called **Stationary Points**.

Stationary points fall into three major categories:

(a) Those in which changes sign from positives through zero to negative. These are called **maximum points.**

(b) Those in which changes sign from negative through zero to positive. These are called **minimum points.**

(c) Those in which the sign of is not changed in the immediate neighborhood of the stationary points. These are called **points of inflexion.**

The terms maximum and minimum points are used in the local sense and not in the absolute sense.

Fig. 10.12 shows a part of the curve y = *f* (x). There is a minimum at x = *b*

At x = b, f1 (b) =0

At x = b^{–}, f^{1} (b^{–}) <0

At x = b^{+, }f^{1} (b^{+}) <0

So a pint on a curve is a minimum at x = a.

If: (i) *f*^{’} (a) = 0 (ii) *f*^{’ }(a^{–}) <0

(iii) *f*^{’}(a^{+})>0

shows part of the curve y= f(x). We observe that

*f*^{’}(c) = 0

*f*^{’}(c^{–}) < 0

*f*^{’}(c^{+}) < 0

The point *x = c* is a point of inflexion. Similarly, fig. 10.13(b) shows parts of the curve y =* f* (x)

We observe that

*f*^{’}(d) = 0

*f*^{’}(d^{–}) >0

*f ’*(d^{+}) >0

The point *x = d* is a point of inflexion.

A maximum point, a minimum point and a point of inflexion are all stationary points. Both maximum and minimum points are called turning points. A point of inflexion however, is not a turning point.

**Example**

Find the stationary points in each of the following curves whose equations are:

(a) y = __x ^{3}__+ x

^{2 }-3x + 4

3

(b) y = __x ^{4}__ +

__4__x

^{3}– 2x

^{2}– 16x + 1

4 3

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