Maxima and Minima two independent variable

MAXIMA AND MINIMA
Of Functions of Two Independent Variables


Definitions

Definition: Let f (x, y) be any function of two independent variables. Sup-
pose f (x, y) is continuous for all values in the neighborhood of their values
a and b respectively. Then f (a, b) is said to be:

1. Maximum Value
f (a, b) is said to be a maximum value of f (x, y) if:

f (a + h, b + k) < f (a, b)

for all sufficiently small independent values of h and k.

2. Minimum Value
f (a, b) is said to be a minimum value of f (x, y) if:

f (a + h, b + k) > f (a, b)

for all sufficiently small independent values of h and k, where h ̸= 0, k ̸= 0.

Necessary Condition
For the existence of a maximum or minimum for f (x, y) at x = a, y = b:
   
∂f ∂f
= 0 and =0
∂x (a,b) ∂y (a,b)



Sufficient Condition
     
∂2f ∂2f ∂2f
Let r = ∂x2
,s= ∂x∂y
,t= ∂y 2
.
(a,b) (a,b) (a,b)



• Case I: If rt − s2 > 0:

1

, – If r < 0, there is a Maximum.
– If r > 0, there is a Minimum.

• Case II: If rt − s2 < 0:

– There is neither maximum nor minimum (This is a Saddle Point).

• Case III: If rt − s2 = 0:

– This case is doubtful and requires further investigation.



Problem 1

Discuss the maximum or minimum values of:

u = 2a2 xy − 3ax2 y − ay 3 + x3 y + xy 3


Step 1: Necessary Conditions

Differentiating with respect to x:
∂u ∂
= (2a2 xy − 3ax2 y − ay 3 + x3 y + xy 3 )
∂x ∂x
= 2a2 y − 6axy + 3x2 y + y 3
∂u
Now set ∂x
= 0:
2a2 y − 6axy + 3x2 y + y 3 = 0
y(2a2 − 6ax + 3x2 + y 2 ) = 0 . . . (i)

Differentiating with respect to y:
∂u
= 2a2 x − 3ax2 − 3ay 2 + x3 + 3xy 2
∂y
∂u
Now set ∂y
= 0:

2a2 x − 3ax2 − 3ay 2 + x3 + 3xy 2 = 0 . . . (ii)

2