If ` a lt 0` and ` f(x)=e^(ax )+ e^(-ax)` is monotonically decreasing . Find the interval to which x belongs.

Options

A. `f {x:x gt 0 } `
B. `{x:x lt 0}`
C. `{x:x lt 1}`
D. `{x:x lt 1}`

Answer

A

Explanation

We have
`f(x)=e^(ax)+e^(-ax) rArr f(x) =a(e^(ax)-e^(-ax))`
For f(x) to be increasing we must have
`f(x) gt 0 `
`rArr a(e^ax-e^(-ax)) gt 0 `
`rArr a(e^(ax)-e^(-ax)) gt 0`
`rArr e^(-bx)-e^(bx) lt 0 ` where a = -b and `b gt 0 `
`rArr e^(bx)-e^(-bx) gt 0 rArr x lt 0 `
Hence `S= {x: x gt 0 }`