Dear friends,

I could not obtain proof for either one of the following inequalities:

$\displaystyle \mathrm{e}^{rx}\geq x+\frac{\mathrm{ln}(r)+1}{r}\text{ for }x\geq0\text{ and }r>0$

or

$\displaystyle \mathrm{e}^{rx}\geq x+\frac{\mathrm{ln}(\mathrm{e}r)}{r}\text{ for }x\geq0\text{ and }r>0.$

I tried to prove it by using their tangents, but it seems not to be so simple.

Any ideas?

Thanks.