Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

Business Mathematics and Statistics

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B Com : Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

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MAXIMA AND MINIMA 
Increasing and Decreasing Functions A function y = f(x) is said to be an increasing function of x in an interval, say a < x < b, if y increases as x increases. i.e. if a < x1 < x2 < b, then f(x1 ) < f(x2 ).
A function y = f(x) is said to be a decreasing function of x in an interval, say a < x < b, if y decreases as x increases. i.e. if a < x1 < x2 < b, then f(x1 ) > f(x, ).

Sign of the derivative Let f be an increasing function defined in a closed interval [a,b]. Then for any two values x1 and x2 in [a, b] with x1 < x2 ,
we have f (x1 ) < f (x2 ).
∴ f(x1 ) < f(x2 ) and x2 - x > 0

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev
Similarly, if f is decreasing on [a,b] then f '(x) ≤ 0, if the derivative exists.

The converse holds with the additional condition, that f is continuous on [a, b].
 Note Let f be continuous on [a,b] and has derivative at each point of the open interval (a,b), then
(i) If f '(x) > 0 for every x ∈ (a,b), then f is strictly increasing on [a,b]
(ii) If f '(x) < 0 for every x ∈ (a,b), then f is strictly decreasing on [a,b]
(iii) If f '(x) = 0 for every x ∈ (a,b), then f is a constant function on [a,b]
(iv) If f '(x) > 0 for every x ∈ (a,b), then f is increasing on [a,b]
(v) If f '(x) < 0 for every x ∈ (a,b), then f is decreasing on [a,b]

The above results are used to test whether a given function is increasing or decreasing.

 Stationary Value of a Function

A function y = f(x) may neither be an increasing function nor be a decreasing function of x at some point of the interval [a,b]. In such a case, y = f(x) is called stationary at that point. At a stationary point f '(x) = 0 and the tangent is parallel to the x - axis.

Example 1

If y = x- 1/x, prove that y is a strictly increasing function x for all real values of x. (x σ 0)

Solution :
We have y = x -1/x Differentiating with respect to x, we get

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev for all values of x, except x = 0
∴ y is a strictly increasing function for all real values of x. (x ≠ 0)

Example 2 If y = 1+ 1/x , show that y is a strictly decreasing function x for all real values of x. (x σ 0)
Solution :

We have   Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev for all values of x. (x ≠ 0) dx x 2

∴  y is a strictly decreasing function for all real values of x. (x ≠ 0)

Example 3 Find the ranges of values of x in which 2x3 - 9x2 +12x + 4 is strictly increasing and strictly decreasing.

Solution : Let y = 2x3 - 9x2 + 12x + 4
dx /dy = 6x2 - 18x + 12
           = 6(x 2 - 3x + 2)
           = 6(x - 2) (x - 1)
 dx /dy  > 0 when x < 1 or x > 2
x lies outside the interval (1, 2).
 dx /dy   < 0 when 1 < x < 2

∴  The function is strictly increasing outside the interval [1, 2] and strictly decreasing in the interval (1, 2)

Example 4 Find the stationary points and the stationary values of the function f(x) = x3 - 3x2 - 9x + 5. Solution :
Let y = x3 - 3x2 - 9x + 5
   dx/dy = 3x 2 - 6x - 9
At stationary points, dy/dx = 0

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev
The stationary points are obtained
when x = -1 and x = 3
when x = -1, y = (-1)3 - 3(-1)2 - 9(-1) + 5 = 10
when x = 3, y = (3)3 - 3(3)2 - 9(3) + 5 = -22  

∴   The stationary values are 10 and -22 The stationary points are (-1, 10) and (3, -22)

Example 5 For the cost function C = 2000 + 1800x - 75x 2 + x 3 find when the total cost (C) is increasing and when it is decreasing. Also discuss the behaviour of the marginal cost (MC)

Solution : Cost function C = 2000 + 1800x  -75x 2 + x 3
Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev
Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

 

∴ C is increasing for 0 < x < 20 and for x > 30. C is decreasing for 20 < x < 30

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev
Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev
∴ MC is decreasing for x < 25 and increasing for x > 25.

 Maximum and Minimum Values

Let f be a function defined on [a,b] and c an interior point of [a,b] (i.e.) c is in the open interval (a,b). Then
(i) f(c) is said to be a maximum or relative maximum of the function f at x = c if there is a neighbourhood (c-δ , c + δ) of c such that for all x ∈ (c- δ, c + δ) other than c, f(c) > f(x)
(ii) f(c) issaid to be a minimumorrelativeminimumofthe function f at x = c if there is a neighbourhood (c- δ, c + δ) of c such that for all x ∈ (c- δ, c + δ) other than c, f(c) < f(x).
(iii) f(c) is said to be an extreme value of f or extremum at c if it is either a maximum or minimum.

Local and Global Maxima and Minima

Consider the graph (Fig. 4.1) of the function y = f(x).

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev



The function y = f(x) has several maximum and minimum points. At the points V , V , ...V , dy = 0. In fact the function has 1 2 8 dx maxima at V1 , V3 , V5 , V7 and minima at V2 , V4 , V6 , V8 . Note that maximum value at V5 is less than the minimum value at V8 . These maxima and minima are called local or relative maxima and minima. If we consider the part of the curve between A and B then the function has absolute maximum or global maximum at V7 and absolute minimum or global minimum at V2 .

Note By the terminology maximum or minimum we mean local maximum or local minimum respectively.

Criteria for Maxima and Minima

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

Concavity and Convexity
Consider the graph (Fig. 4.2) of the function y = f(x). Let PT be the tangent to the curve y = f(x) at the point P. The curve (or an arc of the curve) which lies above the tangent line PT is said to be concave upward or convex downward.

 

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev
The curve (or an arc of the curve) which lies below the tangent line PT (Fig. 4.3) issaid to be convex upward or concave downward.
Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

 Conditions for Concavity and Convexity.

Let f(x) be twice differentiable. Then the curve y = f(x) is (
i) concave upward on any interval if f "(x) > 0
(ii) convex upward on any interval if f "(x) < 0

 Point of Inflection A point on a curve y = f(x), where the concavity changes from upto down or vice versa is called a Point of Inflection.

For example, in y = x 1/3 (Fig. 4.4) has a point of inflection at x = 0

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

Example 6 Investigate the maxima and minima of the function 2x 3 + 3x 2 - 36x + 10. Solution :
Let y = 2x 3 + 3x 2 - 36x + 10 Differentiating with respect to x, we get

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

Again differentiating (1) with respet to x, we get

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev
∴ It attains maximum at x = -3
∴ Maximum value is y = 2(-3)3 + 3(-3)2 - 36(-3) + 10 = 91 d 2 y
when x =2, dx2 = 12(2) + 6 = 30 > 0
∴ It attains minimum at x = 2
∴ Minimum value is y = 2(2)3 + 3(2)2 - 36(2) + 10 = -34

Example 7
Find the absolute (global) maximum and minimum values of the function f(x) = 3x 5 - 25x 3 + 60x + 1 in the interval [-2, 1]

Solution : Given f(x) = 3x5 - 25x 3 + 60x + 1 f 1 (x) = 15x 4 - 75x 2 + 60

The necessary condition for maximum and minimum is f'(x) = 0

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev
Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev
Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

The maximum value when x = -2
is f(-2) = 3(-2)5 - 25(-2)3 + 60(-2) + 1 = -15
The minimum value when x = -1
is f(-1) = 3(-1)5 - 25(-1)3 + 60(-1) + 1 = -37
The maximum value when x = 1
is f(1) = 3(1)5 - 25(1)3 + 60(1) + 1 = 39

∴ Absolute maximum value = 39. and Absolute minimum value = -37

Example 8
What is the maximum slope of the tangent to the curve y = -x 3 + 3x 2 +9x - 27 and at what point is it?

Solution : We have y = -x 3 + 3x 2 +9x - 27 Differentiating with respect to x, we get

dx/dy = = -3x 2 + 6x +9
∴  Slope of the tangent is -3x 2 +6x + 9
Let M = -3x 2 +6x + 9

Differenating with respect to x, we get
dM/dx = -6x + 6 ------------(1)

Slope is maximum when  Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev
       dM/dx = 0 ⇒ -6x + 6 = 0
                       ⇒ x = 1

Again differentiating (1) with respect to x, we get
Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

∴ Maximum value of
M when x = 1 is M = - 3(1)2 + 6(1)+9 = 12

When x = 1 ; y = - (1)3 +3(1)2 +9(1)- 27 = -16
∴ Maximum slope = 12

The required point is (1, -16)

Example 9
Find the points of inflection of the curve y = 2x 4 - 4x 3 + 3.

Solution

 We have y = 2x 4 -4x 3 + 3

Differentiate with respect to x , we get

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev


∴ points of inflection exist.
when x = 0, y = 2(0)4 - 4(0)3 + 3 = 3
when x = 1, y = 2(1)4 - 4(1)3 + 3 = 1
∴ The points of inflection are (0, 3) and (1, 1)


Example 10
Find the intervals on which the curve f(x) = x 3-6x 2+9x-8 is convex upward and convex downward.

Solution : We have f(x) = x 3 - 6x 2 + 9x - 8

Differentiating with respect o x,

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

 

Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev Maxima And Minima - Differentiation, Business Mathematics & Statistics B Com Notes | EduRev

 

∴ The curve is convex upward in the interval (-∞, 2)
The curve is convex downward in the interval (2, ∞)

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