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Question

If a2x4+b2y4=c6,then maximum value of xy is


A

c2ab

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B

c3ab

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C

c32ab

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D

c32ab

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Solution

The correct option is C

c32ab


The explanation for the correct option

Step 1. Find the maximum value of xy:

Given, a2x4+b2y4=c6

y=c6-a2x4b214

Now,

fx=xy=xc6-a2x4b214=c6x4-a2x8b214

Step 2. Differentiatefx with respect to x

f'x=14c6x4-a2x8b2-344x3c6-8a2x7b2

Put f'x=0

4x3c6-8a2x7b2=0

x4=c62a2

x=±c32214a

Step 3. At x=c32214a, f(x) is maximum and hence

fc32212a=c122a2b2-c124a2b214=c124a2b214=c3ab

Hence, Option ‘C’ is Correct.

Alternative method:

We know that

Arithmetic mean Geometric mean i.e.AMGM

a2x4+b2y42a2x4×b2y412

c62ax2by2

c62abxy2

xy2c62ab

Thus, -c32abxyc32ab

Hence, Option ‘C’ is Correct.


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