Exercise 6.5 - Application of Derivative NCERT Solutions | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
 
Exercise 6.5                                                            Page No: 231 
 
1. Find the maximum and minimum values, if any, of the following functions given by: 
(i)  
(ii)  
(iii)   
(iv)  
 
Solution:  
(i) Given function is: 
  
As,  for all x ? R 
Adding 3 both sides, we get 
  
  
The minimum value of f(x) is 3 when 2x – 1 = 0, which means  
This function does not have a maximum value. 
(ii) Given function is:  
 
Using squaring method for a quadratic equation: 
  
  
=  
Page 2


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
 
Exercise 6.5                                                            Page No: 231 
 
1. Find the maximum and minimum values, if any, of the following functions given by: 
(i)  
(ii)  
(iii)   
(iv)  
 
Solution:  
(i) Given function is: 
  
As,  for all x ? R 
Adding 3 both sides, we get 
  
  
The minimum value of f(x) is 3 when 2x – 1 = 0, which means  
This function does not have a maximum value. 
(ii) Given function is:  
 
Using squaring method for a quadratic equation: 
  
  
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
  ……….(i) 
As  for all x ? R 
Subtracting 2 from both sides, 
  
  
Therefore, minimum value of f(x) is -2 and is obtained when  
, that is,  
And this function does not have a maximum value. 
(iii) Given function is:  ……….(i) 
As  for all  R 
Multiplying both sides by  and adding 10 both sides, 
 
  [Using equation (1)] 
Maximum value of f(x) is 10 which is obtained when 
x -1 = 0 which implies x = 1.   
 And minimum value of f(x) does not exist. 
(iv) Given function is:  
At   
At   
Page 3


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
 
Exercise 6.5                                                            Page No: 231 
 
1. Find the maximum and minimum values, if any, of the following functions given by: 
(i)  
(ii)  
(iii)   
(iv)  
 
Solution:  
(i) Given function is: 
  
As,  for all x ? R 
Adding 3 both sides, we get 
  
  
The minimum value of f(x) is 3 when 2x – 1 = 0, which means  
This function does not have a maximum value. 
(ii) Given function is:  
 
Using squaring method for a quadratic equation: 
  
  
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
  ……….(i) 
As  for all x ? R 
Subtracting 2 from both sides, 
  
  
Therefore, minimum value of f(x) is -2 and is obtained when  
, that is,  
And this function does not have a maximum value. 
(iii) Given function is:  ……….(i) 
As  for all  R 
Multiplying both sides by  and adding 10 both sides, 
 
  [Using equation (1)] 
Maximum value of f(x) is 10 which is obtained when 
x -1 = 0 which implies x = 1.   
 And minimum value of f(x) does not exist. 
(iv) Given function is:  
At   
At   
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
Hence, maximum value and minimum value of g(x) do not exist. 
2. Find the maximum and minimum values, if any, of the following functions given by: 
(i)  
(ii)  
(iii)  
(iv)  
(v)  
Solution: (i) Given function is:  ……….(1) 
 
As  for all  R 
Subtracting 1 from both sides,  
  
Therefore, minimum value of f(x) is -1 which is obtained when x + 2 = 0 or x = -2. 
From equation (1), maximum value of  hence it does not exist. 
(ii) Given function is:  
As  for all  R 
Multiplying by  both sides and adding 3 both sides, 
 
 
Maximum value of g(x) is 3 which is obtained when x + 1 = 0 or x = -1.  
From equation (1), minimum value of  , does not exist. 
 
 
Page 4


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
 
Exercise 6.5                                                            Page No: 231 
 
1. Find the maximum and minimum values, if any, of the following functions given by: 
(i)  
(ii)  
(iii)   
(iv)  
 
Solution:  
(i) Given function is: 
  
As,  for all x ? R 
Adding 3 both sides, we get 
  
  
The minimum value of f(x) is 3 when 2x – 1 = 0, which means  
This function does not have a maximum value. 
(ii) Given function is:  
 
Using squaring method for a quadratic equation: 
  
  
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
  ……….(i) 
As  for all x ? R 
Subtracting 2 from both sides, 
  
  
Therefore, minimum value of f(x) is -2 and is obtained when  
, that is,  
And this function does not have a maximum value. 
(iii) Given function is:  ……….(i) 
As  for all  R 
Multiplying both sides by  and adding 10 both sides, 
 
  [Using equation (1)] 
Maximum value of f(x) is 10 which is obtained when 
x -1 = 0 which implies x = 1.   
 And minimum value of f(x) does not exist. 
(iv) Given function is:  
At   
At   
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
Hence, maximum value and minimum value of g(x) do not exist. 
2. Find the maximum and minimum values, if any, of the following functions given by: 
(i)  
(ii)  
(iii)  
(iv)  
(v)  
Solution: (i) Given function is:  ……….(1) 
 
As  for all  R 
Subtracting 1 from both sides,  
  
Therefore, minimum value of f(x) is -1 which is obtained when x + 2 = 0 or x = -2. 
From equation (1), maximum value of  hence it does not exist. 
(ii) Given function is:  
As  for all  R 
Multiplying by  both sides and adding 3 both sides, 
 
 
Maximum value of g(x) is 3 which is obtained when x + 1 = 0 or x = -1.  
From equation (1), minimum value of  , does not exist. 
 
 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
(iii) Given function is:  ……….(i) 
As  for all  R 
Adding 5 to all sides,  
  
Therefore, minimum value of  is 4 and maximum value is 6. 
(iv) Given function is:  
As  for all  R 
Adding 3 to all sides,  
  
Therefore, minimum value of f(x) is 2 and maximum value is 4. 
(v) Given function is:  ……….(i) 
As  
Adding 1 to both sides,  
  
Therefore, neither minimum value not maximum value of h(x) exists. 
 
3. Find the local maxima and local minima, if any, of the following functions. Find also the 
local maximum and the local minimum values, as the case may be: 
(i)  
(ii)  
(iii)  
(iv)  
Page 5


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
 
Exercise 6.5                                                            Page No: 231 
 
1. Find the maximum and minimum values, if any, of the following functions given by: 
(i)  
(ii)  
(iii)   
(iv)  
 
Solution:  
(i) Given function is: 
  
As,  for all x ? R 
Adding 3 both sides, we get 
  
  
The minimum value of f(x) is 3 when 2x – 1 = 0, which means  
This function does not have a maximum value. 
(ii) Given function is:  
 
Using squaring method for a quadratic equation: 
  
  
=  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
  ……….(i) 
As  for all x ? R 
Subtracting 2 from both sides, 
  
  
Therefore, minimum value of f(x) is -2 and is obtained when  
, that is,  
And this function does not have a maximum value. 
(iii) Given function is:  ……….(i) 
As  for all  R 
Multiplying both sides by  and adding 10 both sides, 
 
  [Using equation (1)] 
Maximum value of f(x) is 10 which is obtained when 
x -1 = 0 which implies x = 1.   
 And minimum value of f(x) does not exist. 
(iv) Given function is:  
At   
At   
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
Hence, maximum value and minimum value of g(x) do not exist. 
2. Find the maximum and minimum values, if any, of the following functions given by: 
(i)  
(ii)  
(iii)  
(iv)  
(v)  
Solution: (i) Given function is:  ……….(1) 
 
As  for all  R 
Subtracting 1 from both sides,  
  
Therefore, minimum value of f(x) is -1 which is obtained when x + 2 = 0 or x = -2. 
From equation (1), maximum value of  hence it does not exist. 
(ii) Given function is:  
As  for all  R 
Multiplying by  both sides and adding 3 both sides, 
 
 
Maximum value of g(x) is 3 which is obtained when x + 1 = 0 or x = -1.  
From equation (1), minimum value of  , does not exist. 
 
 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
(iii) Given function is:  ……….(i) 
As  for all  R 
Adding 5 to all sides,  
  
Therefore, minimum value of  is 4 and maximum value is 6. 
(iv) Given function is:  
As  for all  R 
Adding 3 to all sides,  
  
Therefore, minimum value of f(x) is 2 and maximum value is 4. 
(v) Given function is:  ……….(i) 
As  
Adding 1 to both sides,  
  
Therefore, neither minimum value not maximum value of h(x) exists. 
 
3. Find the local maxima and local minima, if any, of the following functions. Find also the 
local maximum and the local minimum values, as the case may be: 
(i)  
(ii)  
(iii)  
(iv)  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
(v)  
(vi)  
(vii)  
(viii)  
Solution: (i) Given function is:  
  and  
Now  
  [Turning point] 
Again, when x = 0,  [Positive] 
Therefore, x=0, is a point of local minima and local minimum value =  
(ii) Given function is:  
  and  
Now  
  
  
  
  or  [Turning points] 
Again, when , 
 [Negative] 
  is a point of local maxima and local maximum value