For the function `f(x)=x cos ""1/x, x ge 1 `which one of the following is incorrect ?

Options

A. for at least one x in the interval `[1,oo),f(x +2)-f(x) lt 2 `
B. `underset(x rarr oo)lim f' (x)=1`
C. for all x in the interval `[1,oo),f(x+2)-f(x) gt 2 `
D. f(x) is strictly decreasing in the interval `[1,oo)`

Answer

A

Explanation

We have
`f(x)=x cos""1/x, c le 1 `
`rArr f(x) = cos ""1/x+1/x sin""1/x and f''(x)=-1/x^3 cos""1/x`
`rArr underset(x rarr oo) f(x)=1 and f''(x) lt 0 " for all" x ge 1 `
`rArr f(x)` is steictly decreasing in the interval [1,oo)
Let g(x) =f(x+2)-f(x) Then
g(x) =f' f(x+2)-f (x) Then
`g(x)=f'(x+2) -f'(x+2)-f(x) lt 0 " "[because f(x)` is strictly decreasing ]
`rArr` g(x) is stricly decreasing on `[1,oo)`
Now
`underset(x rarr o)lim g(x) = underset(x rarr oo ) lim f(x+2)-f(x)`
`underset(x rarr oo)lim (x+2) cos "" 1/(x+2)-x cos""1/x`
`= underset(x rarr oo) lim x (cos""1/(x+2)-cos""1/x)+ 2 cos ""1/(x+2)`
`= underset(x rarr oo)2x sin (x+1)/(x^2+2x)sin""(1)/(x^2+2x)+2 cos""(1)/(x+2)`
`underset( x rarr oo )lim (sin"" ((x+1)/(x^2+2x)))/(((x+1)/(x^2+2x)))xx sin ((1)/(x^2+2x))/(((1)/(x^2+2x))) xx (x(x+1))/(x^2+2x^2)^2+2 cos"" 1/(x+2)`
`=2xx1xx1xx0+2=2`
As g(x) is decresing on `[1 , oo)`
`rArr f(x+2)-f(x) gt 2 "for all " x in [1,oo)`
`rArr f(x+2)-f(x) lt 2 "for all" x in [1,oo)`
Hence option (a ) is incorrect .