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The maximum value of x^4 y^4 when a^2 x^4+b^2 y^4=c^6 is

Options

  1. Ac¹² 16 a^4 b^4
  2. Bc¹² 4 a^2 b^2
  3. Cc^6 (a+b)¹²
  4. Dc^6 a^4+b^4

Correct answer

B. c¹² 4 a^2 b^2

Step-by-step solution

aligned & Let f(x)=x^4 y^4 & And a^2 x^4+b^2 y^4=c^6 & y^4= 1 b^2 (c^6-a^2 x^4 ) & f(x)= x ^4 1 b^2 (c^6-a^2 x^4 ) & = 1 b^2 x^4 (c^6-a^2 x^4 ) & f^ (x)= 1 b^2 (4 x^ 3 c^6 -8 a^2 x^7 ) & aligned for extrem point: aligned & f^ ( x )=0 4 x^3 (c^6-2 a^2 x^4 )=0 & x=0 or x^4= c^6 2 a^2 aligned For maxima : x= c^6 2 a^2 The maximum value is : aligned & f (x= ( c^6 2 a^2 )^ 1 4 ) & = c^6 2 a^2 1 b^2 [c^6-a^2 c^6 2 a^2 ] & = c^6 2 a^2 b^2 c^6 2 = c¹² 4 a^2 b^2 aligned

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