**Summary**

- If
*y = f(x)*is an increasing function then - If
*y = f(x)*is a decreasing function then - Stationary Points are obtained by solving the equation
- 3 types of stationary points:

i) Maximum Point

ii) Minimum Point

iii) Point of Inflexion

#### Increasing Function

Refer to Fig 1, consider a point A on the curve* y = f(x)*. A tangent is drawn at A and the gradient of the tangent at A is .

In this situation when x increases the function *f(x)* also increases so positive. Hence, it is known as an increasing function.

If *y = f(x)* is an increasing function then .

#### Decreasing Function

Refer to Fig 2, A tangent is drawn at A and the gradient of the tangent at A is .

In this situation when x increases the function *f(x)* decreases so is negative. Hence, it is known as a decreasing function.

If *y = f(x)* is a decreasing function then .

#### Example #1

Q. Find the range of values of x such that is an increasing function.

*Solution:*

As *f(x)* is an increasing function hence .

So,

*Ans*

#### Stationary Points

Refer to Fig 3, consider a graph of *y = f(x)*. Notice the 4 main situations in the diagram.

i) Along AB, gradient of the curve is increasing, therefore .

ii) On reaching point B, the tangent at B becomes parallel to the x axis. Hence, at B .

iii) After passing B, the gradient of the curve becomes negative, therefore .

iv) As the curve now reaches C, at C is again 0.

To conclude, point B and C are special points where the tangents being parallel to the x axis, is 0.

Stationary Points are obtained by solving the equation

#### Example #2

Q. Find the stationary values of

*Solution:*

Find

We know that at stationary points .

Hence:

By substituting both values of x at in *f(x)*, we can find the stationary points.

At

At

* Ans:* Therefore coordinates of the stationary points are &

A stationary point is also called a ‘turning point’ if it is either a maximum point or minimum point.

We are now aware that a curve has a stationary point where .

#### There exist 3 types of stationary points:

**1. Maximum Point**

In Fig 4 , before the stationary point (A), the gradient is positive and immediately after the stationary point (A), is negative . Hence, A is a maximum point.

**2. Minimum Point**

In Fig 5 , before the stationary point (A), the gradient is negative and immediately after the stationary point (A), is positive . Hence, A is a minimum point.

**3. Point of Inflexion**

In Fig 6, points A and B are neither maximum nor minimum. Although at both the points, but it does not change sign when moving through A and B i.e if x is positive before A, it will remain positive after A as well. Similarly if x is negative before B, it will remain negative after B as well. Such points are called ‘Point of inflexion’.

To find the nature of a stationary point, we need to find the second derivative of the function and substitute the stationary points in the equation. If our answer is positive it means that its a minimum point and vice versa.

If we differentiate a function again after finding , it is called a second derivative .

Remember:

#### Example #3

Q. The equation of a curve is

i) Express and in terms of *x*.

ii) Find the coordinates of the two stationary points and determine the nature of each stationary point.

*Solution:*

i)

ii) Equate to 0

Hence:

When: *x = 2*

Since , *x = 2* is a minimum point.

When: *x = 1*

Since , x = 1 is a maximum point.