Question

If `x^2-kx+sin^(-1)(sin4) > 0` for all real `x`, then what is the range of `k`?

Answer 1

`x^2-kx+sin^(-1)(sin4) > 0` for all real `x` if `kin(-4,4)`

Explanation:

As `sin^(-1)(sin4)=4`,

`x^2-kx+sin^(-1)(sin4)` can be written as

`x^2-kx+4`

or `x^2-2xxk/2x+(k/2)^2-(k/2)^2+4`

= `(x-k/2)^2+4-k^2/4`

Hence, as `(x-k/2)^2` is always greater than or equal to `0`

`x^2-kx+4 >0` if `4-k^2/4 > 0`

or `16-k^2 > 0`

or `(4-k)(4+k) > 0`

and this will be so if `kin(-4,4)`