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

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Answer 1 · 2017-07-03T08:43:01

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

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 $kin(-4,4)$

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