Prove that `e^(x) ge 1 +x` and hence `e^(x) +sqrt(1+e^(2x))ge(1+x)+sqrt(2+2x+x^(2)) forall` x in R

Explanation

consider function `f(x) =e^(x)-1-x`
`f(x)=e^(x)-1`
for `xge0,f(x)ge0`
f(x) is increasing function for `xge0`
For `xge0,f(x)gef(0)`
or `e^(x)-1-xge0`
or `e^(x)1+x`
For `xlt0,f(x)lt0`
f(x) is decreasing function for `xlt0`
`e^(x)-1-xge0`
or `e^(x)ge1+x`
Thus `e^(x)1+xforall_(x)`in R.
Now we have to prove that
`e^(x)+sqrt(1+e^(2x))ge(1+x)+sqrt(1+(1+x)^(2))`
Thus for `xlt0,f(X)gt0`
Therefore f(x) is an increasing function
since `e^(x)gex+1`, we have
`f(e^(x))gef(x+1)`
or `e^(x)+sqrt(1+e^(2x))ge(1+x)+sqrt(1+(1+x)^(2)).forall`x in R