Question

3.
х3 + y3 + 3axy find maxima and minima
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Answer 1

Answer:

fx = 3x² -3ay

fy = 3y2 -3ax

r = fxx = 6x

s = fxy = -3a

t = fyy = 6y

To find stationary points

• fx = 0

3x² -3ay = 0

3x² = 3ay

x² = ay

y = x²/a

•. fy = 0

3y² = 3ax

3y2 -3ax = 0

on applying the value of y, we get

3(x²/a)² = 3ax

on solving we get

x = a and

x = 0

The corresponding y values for x = 0 and a are y = 0 and a respectively.

The stationary points are (0,0) and (a,a)

(i) At (0,0)

r = 0

s = -3a

t = 0

∆ = rt-s² (formula)

∆ =0×0-(-3a)²

∆ = -9a²

since ∆<0, This point is a saddle point and has no extreme value

(ii) At (a,a)

r = 6a

s = -3a

t = 6a

∆ = rt-s²

∆ =6a×6a-(-3a)²

∆ = 27a²

* if the value of a>o

and , r > 0

∆ > 0

then The function is Minimum

* if the value of a<o

and , r < 0

∆ > 0

then The function is Maximum

hope it helps you ...