PPT: Maxima & Minima | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


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