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Maxima & Minima for Function of 
Single Variable
? A function is maximum at x = c, if f(x) = f(c), ? x
? A function is minimum at x = c, if f(x) = f(c), ? x
Steps to find maxima or minima :-
1. Find f’(x)
2. Equate f’(x) = 0 for obtaining the stationary points
3. At each stationary points find f”(x)
a) If f”(x
0
) > 0 then f(x) has minima at x = x
0
b) If f”(x
0
) < 0 then f(x) has maxima at x = x
0
c) If f”(x
0
) = 0 then f(x) has no extreme at x = x
0
, called critical point
Page 2


Maxima & Minima for Function of 
Single Variable
? A function is maximum at x = c, if f(x) = f(c), ? x
? A function is minimum at x = c, if f(x) = f(c), ? x
Steps to find maxima or minima :-
1. Find f’(x)
2. Equate f’(x) = 0 for obtaining the stationary points
3. At each stationary points find f”(x)
a) If f”(x
0
) > 0 then f(x) has minima at x = x
0
b) If f”(x
0
) < 0 then f(x) has maxima at x = x
0
c) If f”(x
0
) = 0 then f(x) has no extreme at x = x
0
, called critical point
Maxima & Minima for Function of 
Single Variable
? Question :- ?? ?? = 2 ?? 3
- 3 ?? 2
- 36?? + 10 has a maximum value at x = ________.
? Solution:- ?? '
?? = 6 ?? 2
- 6?? - 36 = 0
? ?? 2
- ?? - 6 = 0
x = -2    x = 3
?? ''
?? = 12 ?? - 6
At x = -2, ?? ''
?? = 12 - 2 - 6 = -30 < 0 ? maxima at x = -2
At x = 3, ?? ''
?? = 12 3 - 6 = 30 > 0 ? minima at x = 3
Page 3


Maxima & Minima for Function of 
Single Variable
? A function is maximum at x = c, if f(x) = f(c), ? x
? A function is minimum at x = c, if f(x) = f(c), ? x
Steps to find maxima or minima :-
1. Find f’(x)
2. Equate f’(x) = 0 for obtaining the stationary points
3. At each stationary points find f”(x)
a) If f”(x
0
) > 0 then f(x) has minima at x = x
0
b) If f”(x
0
) < 0 then f(x) has maxima at x = x
0
c) If f”(x
0
) = 0 then f(x) has no extreme at x = x
0
, called critical point
Maxima & Minima for Function of 
Single Variable
? Question :- ?? ?? = 2 ?? 3
- 3 ?? 2
- 36?? + 10 has a maximum value at x = ________.
? Solution:- ?? '
?? = 6 ?? 2
- 6?? - 36 = 0
? ?? 2
- ?? - 6 = 0
x = -2    x = 3
?? ''
?? = 12 ?? - 6
At x = -2, ?? ''
?? = 12 - 2 - 6 = -30 < 0 ? maxima at x = -2
At x = 3, ?? ''
?? = 12 3 - 6 = 30 > 0 ? minima at x = 3
Maxima & Minima for Function of Two 
Variables
? Let z = f(x, y)
? Then, ?? =
????
????
, ?? =
????
????
, ?? =
?? 2
?? ?? ?? 2
, ?? =
?? 2
?? ?? ?? ????
, ?? =
?? 2
?? ?? ?? 2
Steps to find maxima or minima :-
1. Find p, q, r, s and t
2. Equate p & q to zero to obtain stationary points
3. At each stationary points find r, s and t
a) If rt – s
2 
> 0, r > 0 then f(x, y) has a minima at that stationary point.
b) If rt – s
2 
> 0, r < 0 then f(x, y) has a maxima at that stationary point.
c) If rt – s
2 
< 0, r > 0 then f(x, y) has no extreme at that stationary point at it is known as 
saddle point.
Page 4


Maxima & Minima for Function of 
Single Variable
? A function is maximum at x = c, if f(x) = f(c), ? x
? A function is minimum at x = c, if f(x) = f(c), ? x
Steps to find maxima or minima :-
1. Find f’(x)
2. Equate f’(x) = 0 for obtaining the stationary points
3. At each stationary points find f”(x)
a) If f”(x
0
) > 0 then f(x) has minima at x = x
0
b) If f”(x
0
) < 0 then f(x) has maxima at x = x
0
c) If f”(x
0
) = 0 then f(x) has no extreme at x = x
0
, called critical point
Maxima & Minima for Function of 
Single Variable
? Question :- ?? ?? = 2 ?? 3
- 3 ?? 2
- 36?? + 10 has a maximum value at x = ________.
? Solution:- ?? '
?? = 6 ?? 2
- 6?? - 36 = 0
? ?? 2
- ?? - 6 = 0
x = -2    x = 3
?? ''
?? = 12 ?? - 6
At x = -2, ?? ''
?? = 12 - 2 - 6 = -30 < 0 ? maxima at x = -2
At x = 3, ?? ''
?? = 12 3 - 6 = 30 > 0 ? minima at x = 3
Maxima & Minima for Function of Two 
Variables
? Let z = f(x, y)
? Then, ?? =
????
????
, ?? =
????
????
, ?? =
?? 2
?? ?? ?? 2
, ?? =
?? 2
?? ?? ?? ????
, ?? =
?? 2
?? ?? ?? 2
Steps to find maxima or minima :-
1. Find p, q, r, s and t
2. Equate p & q to zero to obtain stationary points
3. At each stationary points find r, s and t
a) If rt – s
2 
> 0, r > 0 then f(x, y) has a minima at that stationary point.
b) If rt – s
2 
> 0, r < 0 then f(x, y) has a maxima at that stationary point.
c) If rt – s
2 
< 0, r > 0 then f(x, y) has no extreme at that stationary point at it is known as 
saddle point.
Maxima & Minima for Function of Two 
Variables
? ?? ?? , ?? = ?? 3
- 3??
2
+ 4??
2
+ 6 has a minimum value at x = ________.
? Solution:-
? ?? =
????
????
= 3??
2
- 6?? = 0 ? x = 0, 2
? q =
????
?? ?? = 8?? = 0 ? y = 0
? ?? =
?? 2
?? ????
2
= 6 ?? - 6 , ?? =
?? 2
?? ???? ????
= 0 , ?? =
?? 2
?? ?? ?? 2
= 8
? At (0, 0) r = -6 < 0, s = 0, t = 8, rt – s
2 
< 0 ? saddle point
? At (2, 0) r = 6 > 0, s = 0, t = 8, rt – s
2 
> 0 ? point of minima at x = 2
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FAQs on PPT: Maxima & Minima - Engineering Mathematics - Civil Engineering (CE)

1. What is the concept of maxima and minima in mathematics?
Ans. Maxima and minima are fundamental concepts in mathematics that refer to the highest and lowest values of a function or a set of data points. In calculus, maxima and minima are often associated with finding the peak and valley points of a function, where the function's derivative is zero or undefined.
2. How are maxima and minima useful in optimization problems?
Ans. Maxima and minima play a crucial role in optimization problems, where the goal is to find the best possible solution. In such problems, maxima represent the highest achievable value, while minima represent the lowest achievable value. By determining the maxima and minima of a function, we can identify the optimal points that satisfy certain constraints or criteria.
3. Are there different types of maxima and minima?
Ans. Yes, there are different types of maxima and minima. The two main types are absolute maxima and minima, which are the highest and lowest points of a function over its entire domain. Additionally, there can be local maxima and minima, where the function reaches a peak or valley within a specific interval, but not necessarily the highest or lowest overall.
4. How can we find the maxima and minima of a function?
Ans. To find the maxima and minima of a function, we can use various mathematical techniques such as differentiation, critical point analysis, and the first or second derivative tests. By taking the derivative of the function and setting it equal to zero, we can find the critical points where the maxima and minima occur. Additional tests can then be applied to determine whether these critical points correspond to maximum or minimum values.
5. Can maxima and minima be used in real-life applications?
Ans. Yes, the concept of maxima and minima is extensively used in various real-life applications. For example, in economics, maxima and minima help in determining the optimal production level to maximize profits or minimize costs. Similarly, in engineering, maxima and minima aid in designing structures with maximum strength or minimum material usage. Optimization problems in logistics, scheduling, and resource allocation also rely on finding maxima and minima to achieve the best possible outcomes.
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