# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

19,262 questions
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### Inequality with the variable in both the base and the exponent. [duplicate]

I realised that what I took as terseness of my question, actually made it look like a lazy attempt to get a homework answer. The following is the edited question, hopefully up to the standards of this ...
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### Prove using induction that $n^6 < 3^n$,for all $n > 18$

Prove using induction (or using any other elementary precalculus techniques) that $$n^6 < 3^n, \forall n \geq 19.$$ I have no idea how to do this. Writing the induction step, I get that I need to ...
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### If $1<a<b$, which of the following is larger than the other: $a\sqrt[3]{b^2}$ and $b\sqrt[3]{a^2}$? [closed]

Given that $1<a<b$, how can you determine which is the larger, out of $a\sqrt[3]{b^2}$ and $b\sqrt[3]{a^2}$? Thanks in advance.
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### How to find the smallest integer $k > 0$ such that the following inequality holds

I need to find the smallest integer $k>0$ such that the following inequality holds: $$(1-\frac{1}{365})^k\le\frac{1}{2}$$ The answer is supposedly greater than $200$. How can I find $k$?
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### If $a+b=1$ where $a,b\in\Bbb R^+$ then prove that $a^ab^b+b^aa^b\le1$

If $a+b=1$ such that $a$ and $b$ are positive real numbers, then prove $$a^a b^b+b^a a^b\le1$$ I tried applying Arithmetic and Geometric mean but it doesn't work. I also tried to equate the given ...
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### Neighbourhood set in Graph theory [closed]

Let $G$ be any connected graph with $\Delta(G)$ be maximum degree. If $D \subseteq V(G)$ then how can we say that $\left | \bigcup \limits_{v \in D} N(v) \right | \leq |D| \Delta(G)$.
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### Prove that, if $a>c$ and $b>d$, thus $ab>cd$ [closed]

I would like to ask you a question: how could I prove that, if $a>c$ and $b>d$, thus $ab>cd$? Thank you for help. P.s. I forgot to tell you that $a>0, b>0, c>0, d>0.$
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### Why is $\pi^2$ so close to $10$?

Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to form ...
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### Why $\sin x + \cos x$ is non-decreasing in $0 \le x \le \pi/4$? [closed]

How can I prove that $\sin x + \cos x$ is non-decreasing in $0 \le x \le \pi/4$ ?
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### Prove If $\lim_{x\to x_0 }f(x) > C$ then $f(x) > C$ in a deleted neighborhood of $x_0$ and… [closed]

Prove: If $$\lim_{x\to x_0 }f(x) > C$$ then $f(x) > C$ in a deleted neighborhood of $x_0$. Similarly, if $$\lim_{x\to x_0 }f(x) < C$$ then $f(x) < C$ in a deleted neighborhood of $x_0$.
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### How to solve the inequality $x^3 + 2x^2 - 2x - 1 < 0$? [closed]

What will be the solution to the following inequality? $$x^3 + 2x^2 - 2x - 1 < 0$$ I can't solve this using the table of signs method. Please include an explanation as to how it was solved.
### simple inequality in $R ^ 2$ [closed]
Given that, $D\{(a, b), (c, d)\}$ is the usual distance between $(a, b)$ and $(c, d)$ in $R ^ 2$, $0 < k$ and $(x, y) \in R ^ 2$ Is the following true? \$D\{(x, y), (-k, 0)\} - D\{(x, y), (k, 0)...