If x>0, show that 1 + x + x^2/2 < e^x < 1 + x + x^2/2 e^x
I do the easier inequality, hopefully, the second one is similar.
First we will prove that for .
Define .
Then but for .
Therefore, is increasing for .
But since we have it must mean that for .
Therefore, .
Second define .
Then but for by above.
Therefore, is increasing for .
But since we have it must mean that for .
Therefore,