1.

Which of the following is/are incorrect?

Answer»

let `F:RTOR` such that `f(x)=2X+[(x(x^(2)-1))/(4(x^(4)-x^(2)+1))+1/8]+[x]+sinxcosx` then (where [.] denotes the greatest integer function) `f` is one-one onto
let `f:RtoR` such that `f(x)=(x^(3)+2x^(2))/(sinx+2)` then `f` is one-one onto
let `f:Rto[1,oo)` such that `f(x)=2^(|x-1|^(sqrt(2))` then `f(x)` is one-one ito
let `f:R-{0}toR` such that `f(x)=|x|ln|x|` then`f(x)` is one-one onto

Solution :(A) `(x-1/x)/(4(x^(2)+1/(x^(2))-1))=(x-1/x)/(4((x-1//x)^(2)+1))`
sop, `-1/8 le (x-1/x)/(4((x-1/x)^(2)+1))le 1/8`
`:. [(x(x^(2)-1))/(4(x^(4)-x^(2)+1))+1/8]=0`
So, `f(x)=2x+[x]+1/2sin2x` injective but not onto
(B) `f^(')(x)=-x^(3) cos x +x^(3)(3sinx+2cosx)+4x(sinx+2)`
MANY one onto


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