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Show that 1+xin(x+sqrt(x^2+1))geqsqrt(1+...

Show that `1+xin(x+sqrt(x^2+1))geqsqrt(1+x^2)` for all `xgeq0.`

Text Solution

Verified by Experts

`Let f(x)=1+xlog_(e)x+sqrt(x^(2)+1)-sqrt(1+x^(2))`
`therefore f(x)=log_(e)x+sqrt(cx^(2)+1)+x(1+x)/(sqrt(x^(2)+1)/(x+sqrt(x^(2)+1)))-(x)/sqrt(1+x^(2))`
`=log_(e)(x+sqrt(x^(2)+1))ge0 for xge0`
`therefore` f(x) is an increasing function.
So for `xge0,f(x)gef(0)`
`rArr 1+xlog_(e)(x+sqrt(x^(2)+1))=sqrt(1+x^(2)ge0)`
`rArr 1+xlog_(e)(x+sqrt(x^(2)+1))gesqrt(1+x^(2))`
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