10 questions linked to/from Prove that $e^x\ge x+1$ for all real $x$
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### Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
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### Proof of the inequality $e^x-x>0$ for all $x\in \Bbb R.$

Prove that the following inequality is true for all real numbers $$e^x-x>0.$$
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### Proof that $e^{-x} \ge 1-x$

My aim is to prove that $e^{-x} \geq 1-x$ for any $x \geq 0$. What I found so far is Bernoulli's inequality, which states that $$1+x\leq\left(1+\frac{x}{n}\right)^n\xrightarrow [n\to\infty]{} e^x$$ ...
### For the function $y=\ln(x)/x$: Show that maximum value of y occurs when $x = e\ldots$
For the function $y=\ln(x)/x$: Show that maximum value of $y$ occurs when $x = e$. Using this information, show that $x^e <e^x$ for all positive values of $x$. Two positive integers, $a$ and $b$,...