0
votes
0answers
33 views

hint for investigating positivity/negativity of a function

I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y_{1},y_{2},y_{3}\in\mathbb{R}$ and the following function ...
0
votes
1answer
90 views

An inequality involving norms.

I have to know how we can show the following inequality: $\|u\|_{2}\leq\|u_{0}\|_{2}+\int_{0}^{t}\|u_{t}(t,x)\|_{2}dt$ where $\|u\|_{2}=\Big(\int_{\Omega}u^{2}dx\Big)^{1/2}$, $u_{0}=u(x,0)$, ...
0
votes
2answers
180 views

Strong Induction on Inequalities

I'm asked to indicate which natural numbers $n$ each of the below inequality is true, and then I am required to prove this via induction, but I'm wondering what that means... Strong induction? ...
1
vote
3answers
61 views

What is the maximum value of $\frac{2x}{x + 1} + \frac{x}{x - 1}$, if $x \in \mathbb{R}$ and $x > 1$?

What is the maximum value of $$f(x) = \frac{2x}{x + 1} + \frac{x}{x - 1},$$ if $x \in \mathbb{R}$ and $x > 1$? A 2-D plot of of $f$ for $x \in (\infty, \infty)$ is here. Lastly, note that ...
1
vote
1answer
161 views

What is the minimum value of $\sqrt{\frac{2(x - 1)}{x}} + \frac{x + 1}{x}$, if $x > 1$?

What is the minimum value of $$f(x) = \sqrt{\frac{2(x - 1)}{x}} + \frac{x + 1}{x},$$ if $x \in \mathbb{R}$ and $x > 1$? Note that $f$ has a global minimum value of $$f(1) = 2$$ if we allow $x ...
-2
votes
3answers
66 views

Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum for $0 < x \in \mathbb{R}$?

This question is related to this one. My question here is: Does the function $$f(x) = 1 + \frac{1}{x} + \frac{1}{x^2}$$ have a global minimum for $0 < x \in \mathbb{R}$? Thank you!
2
votes
4answers
111 views

Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?

Does the function $$f(x) = 1 + \frac{1}{x} + \frac{1}{x^2},$$ where $x \in {\mathbb{R} \setminus \{0\}}$, have a global minimum? I tried asking WolframAlpha, but it appears to give an inconsistent ...
0
votes
0answers
74 views

Can we find a function satisfiying the following functional inequation?

Question: I would be interested know if there exist (and construct) a function $f:\mathbb N\times \mathbb N \to (0,\infty )$ such that for all integers $p,q$, $f(p,q)=f(q,p)$ and a constant ...
1
vote
1answer
147 views

Show that $\int_0^1f(x)^3dx\le\left(\int_0^1f(x)dx\right)^2$ [duplicate]

Possible Duplicate: Prove that $\int_0^x f^3 \le \left(\int_0^x f\right)^2$ Let $f$ be a differentiable function on $[0,1]$. $f(0)=0$ and $1\ge f'(x)\ge0$. Show that ...
9
votes
1answer
359 views

Prove that $\int_0^1|f''(x)|dx\ge4.$

Let $f$ be a $C^2$ function on $[0,1]$. $f(0)=f(1)=f'(0)=0,f'(1)=1.$ Prove that $\int_0^1|f''(x)|dx\ge4.$ Also determine all possible $f$ when equality occurs.
2
votes
2answers
76 views

Geometric intuition for the inequality $(f(y) - c) ( y - d ) \geq (f(d) - c) ( f^{-1}(c) - d )$

Good day to everyone. I am interested in the geometric intuition for the following statement: Let $f:\mathbb{R} \mapsto \mathbb{R}$ be a monotonically increasing, invertible function and $c,d \in ...