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Results tagged with real-analysis
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user 40782
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
1
vote
2
answers
228
views
A real root of a cubic equation for a stationary point
Let us consider the quartic polynomial in $x$
\begin{equation}
F(x) = (2 a p +2)x^4+ (6a(1-a)p^2+(6-12a)p-6)x^3
+ p(2(a-2)(a-1)a p^2 + 3(5a^2-9a+2)p +12a-18)x^2
- p^2 ((a-2)(4a^2 …
- 371
0
votes
A real root of a cubic equation for a stationary point
Here I present another proof of the Proposition under consideration (called here as Lemma) which has common points with previous Toni's proof.
Lemma. Let us consider the function
$$F' = 4(2ap+2)x^3+ …
- 371
0
votes
On polynomial equation of fourth order depending on two parameters and bound on a maximal root
The existence of the root $x_{*} > 1$ was proved above.
Now we prove the uniquenes of the root $x_{*} > 1$.
Let us suppose that there exists another root $x_{2,*} > 1$.
Without loss of generality …
- 371
0
votes
1
answer
133
views
On polynomial equation of fourth order depending on two parameters and bound on a maximal root
I would like to apologize in advance for a too technical question. Let us consider the following fourth order polynomial equation in $x$:
\begin{eqnarray}
F(x) \equiv (2 a p +2)x^4+ (6a(1-a)p …
- 371
4
votes
3
answers
399
views
On some inequality (upper bound) on a function of two variables
There is a problem (of physical origin) which needs an analytical solution or a hint. Let us consider the following real-valued function of two variables
$y (t,a) = 4 \left(1 + \frac{t}{x(t,a)}\right …
- 371