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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

The correct Answer is:
` [ (1)/(2)- (pi)/(2) ( 1+ (pi)/(3)), (sqrt 3)/(2) - (pi)/(6)( 1+ (pi)/(6))]`
Let `f(x) = 1 + x log (x + sqrt(x ^(2) + 1)) - sqrt(1 + x ^(2))`
` therefore f ' (x) = x (( 1+ (x)/( sqrt (x ^(2) + 1))))/( x + sqrt (x ^(2) + 1)) + log (x + sqrt (x ^(2) + 1))`
`- (x)/(sqrt (x ^(2) + 1) ) = (x)/( sqrt (x ^(2) + 1) ) + log ( x + sqrt (x ^(2) + 1 )) - (x)/(sqrt (x ^(2) + 1 ))`
`rArr f ' (x) = log (x +sqrt ( x ^(2) + 1 ))`
` rArr " " f' (x) ge 0 " " [ because log (x + sqrt (x ^(2) + 1 )) ge 0 ]`
` therefore f (x)` is increasing for ` x ge 0`
`rArr " " f (x) ge f (0)`
` rArr 1+ x log (x + sqrt (1 + x ^(2)) - sqrt ( 1 + x^(2)) ge 1 + 0 -1 `
`rArr 1+ x log ( x + sqrt (1 + x ^(2))) ge sqrt (1 + x ^(2)), AA x ge 0`
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