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AP EAMCET2016MathematicsApplication of Derivatives

If A>0, B>0 and A+B= 3 , then the maximum value of A B is

Options

  1. A1 3
  2. B1 3
  3. C1 2
  4. D3

Correct answer

B. 1 3

Step-by-step solution

Given, A+B= 3 Let y= A By= A ( 3 -A ) Differentiating w.r.t A, we get d y d A = ^2 A ( 3 -A )- ^2 ( 3 -A ) A For maxima or minima, d y d A =0 aligned & ^2 A ( 3 -A )- ^2 ( 3 -A ) A=0 & ^2 A ( 3 -A )= A ^2 ( 3 -A ) & (1+ ^2 A ) ( 3 -A ) & = A [1+ ^2 A ( 3 -A ) ] & ( 3 -A )+ ^2 A ( 3 -A ) & = A+ A ^2 ( 3 -A ) & [ ( 3 -A )- A ] & [1- A ( 3 -A ) ]=0 & ( 3 -A )= A & 2 -A=A 2 A= 2 & A= 6 & A=B= 6 aligned Maximum value of A B= 2 6 aligned & = ( 6 )^2 & = ( 1 3 )^2= 1 3 aligned

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