If f(x) =∫(x8+4)/(x4-2x2 +2)dx andf(0)=0 then
  1. F(x) is an odd function
  2. F(x) has range R
  3. F(x) has at least one real root
  4. F(x) is a monotonic function
Kindly explain to me how all the options are correct

Dear Student,
Please find below the solution to the asked query:

fx=x8+4x4-2x2+2dx=x4-2x2+2x4+2x2+2x4-2x2+2dx=x4+2x2+2dx=x4dx+2x2dx+2dxfx=x55+2x33+2x+cAs, f0=0055+2033+20+c=0c=0So, fx=x55+2x33+2xAs, f-x=-x55+2-x33+2-x=-x55+2x33+2xf-x=fxSo, fx is odd function.As, fx is polynomial.So, the range of fx is R.As, for x=0, fx=0x=0 is one of the root of fx.So, fx has one real root.As, f'x=5x45+2×3x23+2=x4+2x2+2>0 xRf'x>0 xRfx is strictly increasing function.So, fx is a monotonic function.Hence, the corrections are A,B,C and D.

Hope this information will clear your doubts about the topic.
Regards

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