Linked Questions

81
votes
26answers
17k views

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 ...
6
votes
7answers
1k views

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 $$e^x-x>0.$$
4
votes
7answers
1k views

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 $$1+x\leq\left(1+\frac{x}{n}\right)^n\xrightarrow [n\to\infty]{} e^x$$ ...
2
votes
5answers
1k views

Prove that $ex \leq e^x$ for all $x \in \mathbb{R}$

This is easy to prove for negative $x$ but what about positive $x$? Should I use MVT?
2
votes
6answers
758 views

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 $b$,...
0
votes
4answers
133 views

Prove that $e^x>x+1 \forall x\ne 0$ [duplicate]

I need to prove that $e^x>x+1 \forall x\ne 0$. Any ideas of hints about how to begin? I don't have any idea except the graphical way.
2
votes
2answers
87 views

proving by multiplying

Let (*) be $ \left(1+\frac{1}{n}\right)\left(1-\frac{1}{n^2}\right)^n < 1, ∀n∈ℕ$ I have tried many ways to get to (*) We have $\left(1-\frac{1}{n^2}\right)^n <1$ and $-\left(1+\frac{1}{n}\...
2
votes
2answers
63 views

Show that $\left(kx-1\right)e^{kx}>-1$ holds $\forall k\in\mathbb{N}^*, \forall x\in\mathbb{R}^*$

How to show that: $\forall k\in\mathbb{N}^*$ and $\forall x\in\mathbb{R}^*$, the inequality $\left(kx-1\right)e^{kx}>-1$ holds. Thank you for your help.
0
votes
2answers
66 views

How to prove that $er^2 \leq e^{r^2}$ for all $r \in \mathbb{R}$

I'm trying to prove the function $f:\mathbb{R}^2 \to \mathbb{R}$ defined as $$f(x,y)=(x^2+y^2)e^{-(x^2+y^2)}$$ attains a maximum at every point of the unit circle. The determinant of the hessian ...
0
votes
1answer
33 views

Sum of exponential function inequality

For any positive a and n, it seems this inequality holds $$ \sum\limits_{t=n+1}^\infty e^{-at} \leq \frac{1}{a}e^{-an} $$ How can I prove this inequality and does this holds for negative a ?