Engineering Mathematics | Maxima and Minima

Hope Foundation’s International Institute of Information Technology, I²IT, P-14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411 057
Tel - +91 20 22933441 / 2 / 3 | Website - www.isquareit.edu.in ; Email - info@isquareit.edu.in
https://www.quora.com/What-is-the-difference-between-local-minima-maxima-and-absolute-minima-maxima

Engineering Mathematics-I
Maxima and Minima
Prepared by : Prof. Rupali Yeole

Maxima and Minima of Functions of Two
Independent Variables
• Let be a function of two independent variables
and , which is continuous for all values of and in
the neighborhood of i.e. be a point in its
neighborhood which lies inside the region .
),( yxf x
yxy
 kbha  , ba,
R

• The point is called a point of relative minimum,
if for all
Then is called the relative minimum value.
• The point is called a point of relative maximum,
if for all
Then is called the relative minimum value.
),(),( kbhafbaf  kh,
),( baf
),(),( kbhafbaf  kh,
),( baf
 ba,
 ba,

• Stationary point: The point at which function is either
maximum or minimum is known as stationary point.
• Extreme Value: The value of the function at stationary point is
known as extreme value of the function .),( yxf

Working Rule
To determine the maxima and minima (extreme values) of a
function
• Step I : Solve
simultaneously for &
• Step II: Obtain the values of
),( yxf
0&0 





y
f
x
f
x y
2
22
2
2
,,
y
f
t
yx
f
s
x
f
r










• Step III:
1) If & at , then is maximum at
& maximum value of the function is .
2) If & at , then is maximum at
& maximum value of the function is .
3) If at , then is neither maximum nor
minimum at .Such point is called Saddle Point.
4) If at , then no conclusion can be made about
the extreme values of & further investigation is
required.
02
 srt 0r  ba, ),( yxf
 ba,
),( baf
02
 srt 0r  ba, ),( yxf
 ba,
),( baf
02
 srt
 ba, ),( yxf
 ba,
02
 srt
 ba,
),( yxf

Example 1
Q.1 Discuss the maxima and Minima of the function
Answer: Let
Step I: For extreme values
12622
 xyx
  126, 22
 xyxyxf
0,02
0&
3
0)3(2,062
0









yy
y
f
x
xx
x
f

Therefore stationary point is .
• Step II:
 0,3
.2
,0
,2
2
2
2
2
2












y
f
t
yx
f
s
x
f
r

• Step III: At
Hence is maximum at
 0,3
02&
040222


r
srt
     
3
123603
22
min

f
 0,3),( yxf

THANK YOU
For further details please contact
RUPALI YEOLE
Department of Applied Sciences & Engineering
Hope Foundation’s
International Institute of Information Technology, I²IT
P-14, Rajiv Gandhi Infotech Park, MIDC Phase 1, Hinjawadi, Pune – 411 057
www.isquareit.edu.in
Phone : +91 20 22933441 / 2 / 3
rupaliy@isquareit.edu.in | info@isquareit.edu.in

Engineering Mathematics | Maxima and Minima

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