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KCET2022MathematicsApplication of Derivatives

The function f(x)=4 ^3 x-6 ^2 x+12 x+100 is strictly

Options

  1. Adecreasing in [ - 2 , 2 ]
  2. Bdecreasing in [0, 2 ]
  3. Cincreasing in ( , 3 2 )
  4. Ddecreasing in ( 2 , )

Correct answer

D. decreasing in ( 2 , )

Step-by-step solution

Given, f(x)=4 ^3 x-6 ^2 x+12 x+100 aligned f^ (x) & =12 ^2 x x-12 x x+12 x & =12 x [ ^2 x- x+1 ] & =12 x [ ^2 x+(1- x) ] aligned We know, 1- x 0 and ^2 x 0 ^2 x+1- x>0 Therefore, f^ (x)>0 , when x>0 x ( - 2 , 2 ) So, f(x) is increasing when x ( - 2 , 2 ) and f^ (x) < 0 , when x < 0 x ( 2 , 3 2 ) So, f(x) is decreasing when x ( 2 , 3 2 ) Hence, f(x) is decreasing in ( 2 , ) .

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