The question is to prove the inequality $e^x\le e^{x^2} + x$. I tried the Taylor expansion like ${e^x} = 1 + x + \frac{{{x^2}}}{{2!}} + \frac{{{x^3}}}{{3!}} + ...$ and $x + {e^{{x^2}}} = 1 + x + \frac{{{x^4}}}{{2!}} + \frac{{{x^6}}}{{3!}} + ...$ but cannot see anything useful out of this. Anyone can provide some help? Thank you.
52673
Does anybody have a simple proof this inequality  $$e^x\le x+e^{x^2}.$$  Thanks.
80433
By expanding $e^x$ into a series $\sum e^x$ prove that $$\forall x \in \mathbb{R}, x \ge 0 \implies e^{x-1} \ge x$$  Also show when this inequality becomes equality.  I'm not really sure how to attack this problem at all, any tips would be very welcome.
43654
Does anyone provide proof  of $1+x\leq e^x$ for all $x$?  What is the minimum $a(&gt;0)$  such that $1+x\leq a^x$ for all x?
121784
I looking for help with proving the following inequality. Any relevant logarithmic identities would be great. Tried differentiating and taking limits and I'm lost as to how to approach this.   $$\frac 1{x+1}&lt;\ln\left(\frac{x+1}x\right)&lt;0.5\left(\frac 1x+\frac 1{x+1}\right)$$
91951
Any hints on how I can prove the inequality:     $$|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$$   Thank you.
87077
Suppose $\mu(X)&lt;\infty$. Show that if $\ f\in\ L^q(X,\mu)$ for some $1\le q&lt;\infty$ then $\ f\in\ L^r(X,\mu)$ for $1\le r&lt;q$   In the solution the assistant wrote this:  If $1\le r&lt;q$ then $q/r\ge 1$. Let p be the dual exponent to $q/r$ (i.e. $\frac{1}{q/r}+\frac{1}{p}=1)$  then by Hölder Inequality  $\int_{X}|f|^rd\mu\le\underbrace{(\int_{X}|f|^qd\mu)^{r/q}}_{??????}(\int_X1^pd\mu)^{1/p}&lt;\infty$  for me the middle part doesn't make sense (it is not the hölder inequality) can somebody show it ?  instead of this can't we say (only in this case) that   $\int_{X}|f|^rd\mu\le (\int_{X}|f|d\mu)^r$ then  $(\int_{X}|f|d\mu)^r\le ((\int_{X}|f|^qd\mu)^{1/q}(\int_X1^pd\mu)^{1/p})^r&lt;\infty$  ?
72147
Prove $\log(1+x) \lt x$ for $x\gt0$  my attempt:  I show $e^{x}\gt 1+x$ for $x\gt0$ since  $e^{x}=1+x+\frac{n(n-1)}{2}\frac{x^2}{n^2}+...$  so if $x\gt0$ then all terms are positive   so $e^{x}\gt 1+x$ for $x\gt0$  now given $e^{x}\gt 1+x$ for $x\gt0$, can I take $\log$ on both sides and show   $\log(1+x)\lt x$ for $x\gt0$  or do I have to prove firstly that $\exp(x)=e^x$ for $x\gt 0$ then I can take $\log$..
85066
Prove   $$\|x\|_2 \le  \|x\|_1 \le \sqrt n\|x\|_2. \forall x \in \mathbb{R}^n$$  I already proved the first inequality $\|x\|_2 \le \|x\|_1$. Please help me with the second part:  $$\|x\|_1\le \sqrt n \|x\|_2.$$
101472
The statement I'm trying to prove is:     $(x+y)^c\le x^c+y^c$ whenever $0\le x,y$ and $0\le c\le1$.   This comes up in the proof that $|x|_*^c$ is an absolute value whenever $0&lt;c\le1$ and $|x|_*$ is an absolute value over some integral domain $D$. The given statement, however, is simply a question about a real inequality. How do you prove this one?  The case $x=0$ or $y=0$ is trivial, so we can assume that $x,y&gt;0$. Then expanding the definitions we have $e^{c\log(x+y)}\le e^{c\log x}+e^{c\log y}$, but the addition doesn't play well with the exponentials and logs, so I don't see what else can be done. This feels like a convexity result, but I'm not seeing exactly how to make the connection (not to mention that I still haven't proved that $\exp$ is convex and $\log$ is concave, so if I can avoid that I'd prefer to).
70683
If $E(X^2)=1$ and $E(|X|)\ge a &gt;0$, then $P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$.  I can see from the well known inequality $E(|X|) \le E(|X|^2)^{1/2}$ that it must be the case that $a\le 1$. But what to do next I'm not sure.
67285
How does one prove:   $$ E(|X+Y|^r)\leq c_r\bigg[E(|X|^r)+E(|Y|^r)\bigg] $$   where $c_r=1$ if $0&lt;r\leq 1$ and $2^{r-1}$ if $r&gt;1$?   This is a classic result whose proof I once knew but have since forgotten. Can someone please explain so I could learn it again? Thank you.
38566
Prove that $\forall x,y&gt;0, x^x+y^y \geq x^y + y^x$   A friend of mine told me none of the teachers in my school have succeeded in proving this seemingly simple inequality (it was asked at an oral exam last year). I tried it myself, but I made no significant progress.
43493
The Taylor expansion of $\exp(-x)$ is $$e^{−𝑥} = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}  ....$$ What I want to prove is that $$ \frac{x^2}{2!} - \frac{x^3}{3!}  .... $$  is greater than or equal to $0$.  I would really appreciate some help.
123132
I am having difficulty proving the following inequality: $$ \sqrt[n]{n} &lt; 1 + \sqrt{\frac{2}{n}} \quad \text{for all positive integers}\,\,\, n. $$ I am trying to use mathematical induction but I am having trouble going from the left side to the right side (in the induction step) and vice versa.   I also tried to take the power of $n$ on both sides and somehow use the binomial expansion but I am having difficulty with that. Thank you!
74105
Calculus book:     Find all numbers $a$ such that $\forall x, a^x \ge 1+x$    I immediately thought of the inequality $e^x\ge 1+x$ and guessed that the answer was any number $a$ in $[e,\infty)$. After playing around with a graphing app I can see this is definitely true, although I can't explain why.   Why is $e$ specifically the smallest number such that the inequality holds? To my untrained eye, a quick scan of the equation does not give rise to anything involving the constant $e$. Maybe because $1+x$ is only a tangent to the equation $a^x$ when $a=e$? Thanks.
14716
I was asked to prove that:   $$x + \frac{1}{x}\geqslant 2$$  for all values of $ x &gt; 0 $  I tried substituting random numbers into $x$ and I did get the answer greater than $2$. But I have a feeling that this is an unprofessional way of proving this. So how do I prove this inequality?
122721
I need to show that for $x&gt;0$, $1+\frac{x}{2} \ge \sqrt{1+x} \ge 1+\frac{x}{2} - \frac{x^2}{8} $.  I used the geometric/arithmetic mean inequality to show that $1+\frac{x}{2} \ge \sqrt{1+x}$ is indeed true. My issue lies in the second part of this.  In trying to show that $\sqrt{1+x} \ge 1+\frac{x}{2} - \frac{x^2}{8} $, I determined that this is a partial taylor series expansion and I attempted to play around with both inequalities to show that this is true but to no avail.  I would appreciate any help in solving this. Ideally I would like to do this analytically but any solutions using more complicated theorems are welcome.