Linked Questions

4
votes
7answers
534 views

prove:$\left|x+\frac{1}{x}\right|\geq 2$ [duplicate]

prove: $\left|x+\frac{1}{x}\right|\geq 2$ Can I just use $\left|\left(\sqrt{x}\right)^2+\left(\frac{1}{\sqrt{x}}\right)^2\right|\geq 2$ and $\left|\left(\sqrt{-x}\right)^2+\left(\frac{1}{\sqrt{-...
1
vote
5answers
210 views

Prove the inequality $x+\frac{1}{x}\geq 2$? [duplicate]

I'm reading Courant's Calculus. There is an exercise: Prove the inequality $x+\frac{1}{x}\geq 2$. I did the following: For the sake of curiosity, I did: $$x+\frac{1}{x}\geq 2$$ $$x^2+1\geq 2x$$ $$...
3
votes
4answers
115 views

How to easily prove $x+\frac{1}{x} \ge 2 \quad ∀x\in ℝ^+$ [duplicate]

When I tried to solve some certain math problem (an inequation) for pivate exercise purposes, I had to prove that $x+\frac{1}{x} \ge 2 \quad ∀x\in ℝ^+$, I solved it with tools from differential ...
3
votes
5answers
102 views

Show that if $x \ge 1$, then $x+\frac{1}{x}\ge 2$ [duplicate]

So here the problem goes: Show that if $x \ge 1$, then $x+\frac{1}{x}\ge 2$. This is a very interesting word problem that I came across in an old textbook of mine. So I know it's got something to ...
2
votes
4answers
137 views

Proving $\forall x\in \mathbb R$, if $x>0$ then $(x+\frac 1 x \ge 2)$ [duplicate]

Prove $\forall x\in \mathbb R$, if $x>0$ then $(x+\frac 1 x \ge 2)$ I think a proof by contradiction is the easiest in this case, so we have: $\forall x\in \mathbb R :x>0\wedge \neg(x+\frac 1 x ...
4
votes
2answers
199 views

For $x>0$, $x + \frac1x \ge 2$ and equality holds if and only if $x=1$ [duplicate]

Prove that for $x>0$, $x + \frac1x \ge 2$ and equality holds if and only if $x=1$. I have proven that $x+ \frac1x \ge 2$ by re-writing it as $x^2 -2x +1 \ge0$ and factoring to $(x-1)^2\ge0$ which ...
0
votes
2answers
87 views

Prove $\sqrt{x^2+1} + \frac{1}{\sqrt{x^2 +1}}\geq 2$ [duplicate]

I want to show why the last inequality in the problem below $\sqrt{x^2+1} + \frac{1}{\sqrt{x^2 +1}}\geq 2$ holds. It's clear that $x^2\geq 0$ and that equality holds when $x=0$ but how can I clearly ...
0
votes
1answer
106 views

Prove that $x+\frac{1}{x}\geq2$ [duplicate]

To prove $x+\frac{1}{x}\geq2$ where $x$ is a positive real number. This is what i try: $$\text{We need to prove } \hspace{1cm} x+\frac{1}{x}-2\geq 0$$ now,$$\frac{x^2-2x+1}{x}=(x-2)+\frac{1}{x}$$ its ...
1
vote
2answers
37 views

Show that the image of the function $f:(0,\infty)\rightarrow \mathbb{R}$, $f(x)=x+\frac1x$ is the interval $[2,\infty)$. [duplicate]

Show that the image of the function $f:(0,\infty)\rightarrow \mathbb{R}$, $f(x)=x+\dfrac{1}{x}$ is the interval $[2,\infty)$. If $x=1$, then $f(1)=2$. So how can I show that the mage of the function ...
2
votes
0answers
59 views

Show that for all positive x $\in$ $\mathbb{R}$ the inequality $x+\frac {1} {x}\geq 2$ holds. (Hint?) [duplicate]

Show that for all positive x $\in$ $\mathbb{R}$ the inequality $x+\dfrac {1} {x}\geq 2$ holds. Can you give me a hint?
4
votes
5answers
6k views

How to prove that $x^2 +1 \geq 2x$?

I am trying to prove that $x^2 +1 \geq 2x$ without using circular logic (meaning first assuming that this inequality is true and then moving to the $2x$ to the left side and factoring it). Thanks.
7
votes
8answers
732 views

How to prove $\frac xy + \frac yx \ge 2$

I am practicing some homework and I'm stumped. The question asks you to prove that $x \in Z^+, y \in Z^+$ $\frac xy + \frac yx \ge 2$ So I started by proving that this is true when x and y have ...
-1
votes
2answers
1k views

Show that if x>0, x+1/x >= 2. [closed]

How am I to prove this inequality without use of calculus: for any real x>0, x+1/x >= 2 ? Thanks for any help.
7
votes
4answers
100 views

Prove $x + \frac{1}{x} \geq 2$ for $x>0$.

Proof that $x+\frac{1}{x}\geq2$ for $x>0$ Would this be correct? $x*(x+\frac{1}{x}\geq2)$ $x^2+1\geq2x$ $x^2-2x+1\geq2x-2x$ $x^2-2x+1\geq0$ Plug in 1 for x: $(1)^2-2(1)+1\geq0$ $1-2+1\geq0$ ...
0
votes
4answers
77 views

max or min value of a function : y = x + 16 /(x+3) where x > 0, without using differentiation [closed]

Many thanks for your help. I don't know how to solve it. also, How do it determine it is a minimum or maximum. Noted. The student does not know about differentiation.

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