- 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
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 .
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 .
Q. Find the range of values of x such that is an increasing function.
As f(x) is an increasing function hence .
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
Q. Find the stationary values of
We know that at stationary points .
By substituting both values of x at in f(x), we can find the stationary points.
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 .
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.
ii) Equate to 0
When: x = 2
Since , x = 2 is a minimum point.
When: x = 1
Since , x = 1 is a maximum point.