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Manipal MET2020MathematicsApplication of Derivatives

If f^ (x) 0 x R, f^ (3)=0 and g(x)=f ( ^2 x-2 x+4 ), 0 x 2 , then g(x) is increasing in

Options

  1. A(0, 4 )
  2. B( 6 , 3 )
  3. C(0, 3 )
  4. D( 4 , 2 )

Correct answer

D. ( 4 , 2 )

Step-by-step solution

Given, g(x)=f ( ^2 x-2 x+4 ) On differentiating w.r.t. x , we get aligned g^ (x)=f^ ( ^2 x-2 x . & +4) & (2 x ^2 x-2 ^2 x ) aligned =f^ ( ^2 x-2 x+4 ) 2 ^2 x( x-1) f^ (x) 0 x (0, 2 ) Also, 2 ^2 x 0 x (0, 2 ) [ x (0, 2 ) . , given ] But x-1 0 x ( 4 , 2 ) g^ (x) 0 x ( 4 , 2 ) Hence, g(x) is increasing in ( 4 , 2 ) .

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