The function f, defined by ` f(x)=(x^(2))/(2) +In x - 2 cos x ` increases for ` x in `

Options

A. `R^(-)`
B. `R^(+)`
C. `R-{0}`
D. `[1,oo]`

Answer

B

Explanation

`f(x) =(x^(2))/(2) + ln x -2 cos x rArr f.(x)= x+(1)/(x) +2 sin x`
We know that `x +(1)/(2) ge 2 , AA x gt 0 and 2 sin le 2, AA x in R`
`rArr f.(x) gt 0 " for " x gt 0`
Hence f(x) is increasing in `(0, oo)`
When `x lt 0, x +(1)/(x) le -2` but `-2 le 2 sin x le 2, AA x in R`
`rArr f.(0) lt 0` hence f(x) in decreasing in `(-oo, 0)`