Let f(x) be a function such that,f(0)=f′(0)=0 ,f′′(x)=sec4⁡ x+4 then the function is 

 Let f(x) be a function such that,f(0)=f(0)=0 ,f′′(x)=sec4 x+4 then the function is 

  1. A

    log |(sin x)|+13tan3 x+x

  2. B

    23log |(sec x)|+16tan2 x+2x2

  3. C

    log |(cos x)|+16cos2 x+x5

  4. D

    None of the above 

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    Solution:

    Since,f′′(x)=sec4 x+4

    f′′(x)=1+tan2 xsec2 x+4

     f(x)=tan x+tan3 x3+4x+Cf(0)=00=C

     Then, f(x)=tan x+tan3 x3+4x

    f(x)=tan x+tan3  x3+4x=tan x+13tan xsec2x1+4x f(x)=23tan x+13tan xsec2 x+4x f(x)=23log |sec x|+tan2 x6+2x2+d But  f(0)=0d=0

     Then, f(x)=23log |sec x|+16tan2 x+2x2

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