Tagged Questions
1
vote
2answers
65 views
A simple proof about $e^x$?
Do you guys think this is correct? I am trying to prove that there is no single-term polynomial function (oxymoron, I know) $f(x)$ which is always (or at least as x approaches infinity) greater than ...
1
vote
2answers
106 views
Condition for fourth degree polynomial to have all real roots
For what range of values of $a$ will the following fourth degree polynomial have all real roots:
$$x^4 - 2ax^2 + x + a^2 -a = 0$$
0
votes
0answers
58 views
Finding the number of negative and the positive zeros
$$f(x)=x^{4} -3x^{3}+5x^{2}-x-2$$
How to find the number of negative and the positive zeros?
1
vote
3answers
115 views
$x^4 + 4x^3 - 2x^2 - 12x + k$ has 4 real roots. Find the condition on k.
The question is: $f(x) = x^4 + 4x^3 - 2x^2 - 12x + k$ has 4 real roots. What values can k take? Please drop a hint!
3
votes
2answers
92 views
$x^4 + 4rx + 3s = 0$ has no real roots. Relate $r, s$.
It is given that $x^4 + 4rx + 3s = 0$ has no real roots. What can be said about r and s?
a) $r^2 < s^3$
b) $r^2 > s^3$
c) $r^4 < s^3$
d) $r^4 > s^3$
How to even begin??
0
votes
1answer
42 views
Bernoulli polynomials properties
I was reading about Bernoulli polynomials in this article: http://ocw.mit.edu/courses/mathematics/18-100c-analysis-i-spring-2006/projects/silva.pdf and I saw this property:
$$B_n(1−x) = ...
3
votes
2answers
169 views
Show $\;f(x) = x^{20}-70x^3+1\;$ has zero in $\;[0, 1]$
How can we use the Intermediate Value Theorem to show that the function $$\;f(x) = x^{20}-70x^3+1\;$$ has a zero in the interval $\;[0, 1]\,$?
(To use the theorem I need to show that the function ...
2
votes
1answer
62 views
Finding the min and max of $f(x) = \log_{10}x + x^3 - x^2 - 6x + 3$
$$f(x) = \log_{10}x + x^3 - x^2 - 6x + 3$$
$$x > 0$$
How do I find the maxima and the minima of this function?
This is a highscool level problem.
1
vote
2answers
55 views
4-th derivative of $(1+x+x^2) / (1-x+x^2) $ using Taylor polynomial for $1/(1-x)$
Using $n$-th Taylor polynomial for $f_1(x)=\frac{1}{1-x}$ with center in $0$, find $4$-th derivative of $f_2(x)=\frac{1+x+x^2}{1-x+x^2}$ in the point $0$ without calculating it's $1$,$2$ or $3$ ...
4
votes
1answer
47 views
$\inf_{b\in\mathbf{R},c>0,P(x)\in A}{\dfrac{\int_{b}^{b+c}{|P(x)|dx}}{c^{n+1}}}>0$
Let $n\in \mathbb{N}^{+}$, $A=\{f(x)|f(x)=x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0},a_{i}\in \mathbb{R}\} $. Show that
$$\inf_{b\in\mathbf{R},c>0,P(x)\in ...
0
votes
1answer
45 views
How to solve for the coefficients of this polynomial?
For what values of the constants $a$, $b$, $c$, and $d$ does the function $f(x) = ax^3 + bx^2 + cx + d$ satisfy both of the following conditions?
a) $f''(0) = 0$ at the origin
b) a ...
0
votes
2answers
37 views
Algorithm for shifting a curve
I have to following problem that I would like to solve.
I have a vector of coefficients $V = [a_N, \ldots ,a_1, a_0]$ which represents the coefficients of a polynomial $P$, i.e.:
$$
P(x) = a_N x^N + ...
1
vote
1answer
54 views
Equation of the tangent to $4x^2-3xy-y^2=25$ at $(2, -3)$
I have a function $f(x,y)=4x^2-3xy-y^2=25$ and I am trying to find the gradient at $(2,-3)$. First i took the derivative: $$\frac{d}{dx}(f'(x,y))=8x-\left(3y+3x\frac{dy}{dx}\right)-2y\frac{dy}{dx}=0$$
...
2
votes
1answer
71 views
Polynomial function, only 1 solution
I am given the following task:
The graph of the polynomial function $f(x)$ is symmetric to the y-axis. It has exactly one local minimum on the x-axis and an inflection point at $x = 1$. Find the the ...
6
votes
3answers
118 views
Can every real polynomial be factored up to quadratic factors?
I know that you can't factor a real polynomial into $\Pi_{i=1}^N(x-a_i)$ in general. But is it possible to factor every real finite polynomial into this form:
$(\Pi_{i=1}^N a_ix^2 + b_ix + c_i) ...
3
votes
2answers
135 views
Uniform convergence of a sequence of polynomials
If a sequence of polynomials converge uniformly in $\mathbb{R}$ to $f$, is $f$ necessarily a polynomial?
6
votes
0answers
66 views
Monotonic version of Weierstrass approximation theorem
Let $f\in\mathcal{C}^1([0,1])$ be an increasing function over $[0,1]$.
Prove or disprove the existence of a sequence of real polynomials $\{p_n(x)\}_{n\in\mathbb{N}}$ with the properties:
$p_n(x)$ ...
14
votes
3answers
448 views
Shortcut/trick for integrating a factored polynomial?
If I'm integrating a factored polynomial, say
$$\int{x(x+1)(x-2)(x+3)dx},$$
does some shortcut exist that keeps me from having to expand the polynomial? Currently, I'd just do all the multiplication ...
2
votes
1answer
73 views
Evaluate a certain derivative
Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $\{l_1,\dots,l_n\}$ a vector of natural numbers such that $l_1+l_2+\dots+l_n=N$. Let
$$ h_j(x)=\prod_{i\neq j,i=1,\dots, n} ...
1
vote
1answer
55 views
Bernoulli polynomials, Apostol
Define Bernoulli polynomials as:
$P_0(x)=1$, $P'_n(x)=nP_{n-1}(x)$, $\int_0^1P_n(x)=0$ if $n\geq1$
Need to prove that for $n\geq2$ we have
$$\sum_{r=1}^{k-1} r^n= \frac{P_{n+1}(k)-P_{n+1}(0)}{n+1}$$
...
3
votes
1answer
91 views
Integrating an n-th degree polynomial
Integrating a polynomial over a fixed interval is usually very straightforward. However, I can't seem to get very far with an $n$-th degree polynomial:
$$\int_a^b \bigg(\sum_{i=0}^n ...
2
votes
1answer
33 views
Sequence of polynomials (Q2)
Define $P_0(x) = 0$ and for $n > 0, \ P_n(x) = (x \ + \ P_{n-1}^2(x)) / 2$ and $Q_n(x) = P_n(x) - P_{n-1}(x)$. Are all the coefficients of the polynomials $Q_n(x)$ nonnegative?
-3
votes
1answer
56 views
Question 3 on calculus [closed]
Show that the function $f(x)=x^3+5x-5$ increases on $(-\infty,+\infty)$. Hence deduce that the equation $x^3+5x-5=0$ has a unique real solution.
How many real solutions does the equation $x^3+5x-5=0$ ...
1
vote
1answer
48 views
Bernoulli Polynomials
I am having a problem with this question. Can someone help me please.
We are defining a sequence of polynomials such that:
$P_0(x)=1; P_n'(x)=nP_{n-1}(x) \mbox{ and} \int_{0}^1P_n(x)dx=0$
I need to ...
5
votes
2answers
90 views
Prove that $\frac{n-1}{n}>\frac{2a_0a_2}{a_1^2}$
Given that the following equation $$p(x)=a_0x^n+a_1x^{n-1}+...+a_{n-1}x+a_n=0$$ has $n$ distinct real roots. Prove that
$$\frac{n-1}{n}>\frac{2a_0a_2}{a_1^2}$$
3
votes
1answer
76 views
Slope of tangent at…
Can anyone check my math on the following equation? I'm trying to find the slope of the tangent at$ x=3.5 $ for $$ y = 0.000005x^4 - 0.0014x^3 + 0.1007x^2 - 4.4776x + 168.79$$
It's been a very long ...
0
votes
2answers
67 views
Help to show if the function is decreasing for large $l$
I would like to see if
$$
b_l:=4^{-l} \sum_{j=0}^l \frac{\binom{2 l}{2 j} \binom{n}{j}^2}{\binom{2 n}{2 j}}\text{.}
$$
is decreasing when $l$ is large enough say around $10^6$. I dont need any ...
1
vote
1answer
108 views
Intersection of a line with a curve
I have the following question:
Given a line $y=\theta(1-x)$ where $0<x<1$, $0<y<1$ and $0<\theta<1$, I have a collection of curves
$$
y^K=1-(1-x)^K
$$
parametrized by an ...
6
votes
4answers
237 views
Nth derivative of $\tan^m x$
$m$ is positive integer,
$n$ is non-negative integer.
$$f_n(x)=\frac {d^n}{dx^n} (\tan ^m(x))$$
$P_n(x)=f_n(\arctan(x))$
I would like to find the polynomials that are defined as above
...
1
vote
1answer
175 views
maximum of the function at limit
I have a simple question.
Let
$$P(\theta;K) = \left(1-\theta\right)^K\left[\frac{1-(1-\theta)^K-\theta^K}{(1-\theta)^K+\theta^K}-\sum_{i=1}^{\frac{K-1}{2}}\left(\begin{array}{l}
K \\
...
4
votes
0answers
368 views
Gauss-Lucas Theorem (roots of derivatives)
Gauss-Lucas Theorem states:
"Let f be a polynomial and $f'$ the derivative of $f$. Then the theorem states that the $n-1$ roots of $f'$ all lie within the convex hull of the $n$ roots ...
1
vote
2answers
189 views
Integrating Reciprocals of Polynomials
I have seen integrals of the form $$\int \frac{1}{ax+b}dx$$ and $$\int \frac{1}{ax^{2}+bx+c} dx$$
But I cannot see how to integrate reciprocals of higher degree - does there exist a general solution ...
5
votes
1answer
235 views
Antiderivative of Polynomials
I really like how differentiation is introduced for polynomials:
Let $P(t) \in A[t]$ :
$$D_P(t,s) = \frac{P(t) - P(s)}{t-s} \;\; \in A[t,s]$$
and the derivative of $P$ is
$$P'(t) = D_P(t,t).$$
It ...
1
vote
1answer
80 views
Greater than zero?
I need to show that
$$\sum_{i=k^*}^K\binom{K}{i}a^{i-1}(1-a)^{K-i-1}(i-aK)>0$$
given $K\geq k^*$, $0<a<1$ and $K$, $k^*\in\mathbb{Z^+/1}$.
I did some computer simulation and saw that it ...
1
vote
2answers
113 views
Rolle's theorem for showing that $(x-a)^k$ divides $p(x)$ . . .
Have the following I'm stuck on:
Suppose $p(x)=p_0+p_1 x+p_2 x^2+\cdots+p_n x^n$ is a polynomial of degree $n \geq 1$. Show that if $(x-a)^k$ divides $p(x)$ for some $a\in\mathbb R$ and some ...
24
votes
2answers
541 views
Countability of the zero set of a real polynomial
This is the question from my calculus homework: Is it possible for a polynomial $f\colon\, \mathbb{R}^{n}\to \mathbb{R}$ to have a countable zero-set $f^{-1}(\{0\})$?
(By countable I mean countably ...
4
votes
0answers
168 views
Simple formulation, nontrivial problem
There's a problem from calculus I remember:
$$\forall x\ \exists n.\ f^{(n)}(x) = 0 \iff \exists n\ \forall x.\ f^{(n)}(x) = 0\,.$$
Function $f \in C^\infty(\mathbb{R})$, and the notation $f^{(n)}$ ...
9
votes
4answers
189 views
sum of reciprocals of derivative of polynomial at its roots
If $P(x)$ is a polynomial of degree $n > 1$ with only simple roots $a_1,\ldots,a_n$, is it true that $\frac 1{P'(a_1)} + \cdots + \frac 1{P'(a_n)} = 0$, and, if so, what is the proof? I ...
1
vote
1answer
230 views
A problem on Lagrange interpolation polynomials
Based on a previous question, I had the following conjecture and was wondering if anyone knew how to prove it or find a counterexample.
Consider the polynomial $$ ...
2
votes
1answer
374 views
Remainder term of Lagrange Interpolation Polynomial
Suppose $x_0,x_1,\ldots,x_n$ are $n+1$ distinct numbers in the interval $[a,b]$ and $f\in C^{n+1}[a,b]$. Then for each $x$ in $[a,b]$, there is a number $\xi$ in $(a,b)$ such that
$$f(x) = P(x) + ...
4
votes
3answers
205 views
If $f(x)$ and $f(x)-x$ have only one real root, then $f(f(x))-x$ has only one real root.
First edition was: Let $f(x)$ be a polynomial such that $f(x)$ and $f(x)-x$ have only one real root. How to prove, without derivatives, that
$f(f(x))-x$ also has only one real root?
Second edition: ...
2
votes
2answers
122 views
Changing the argument for a higher order derivative
I start with the following:
$$\frac{d^n}{dx^n} \left[(1-x^2)^{n+\alpha-1/2}\right]$$
Which is part of the Rodrigues definition of a Gegenbauer polynomial. Gegenbauer polynomials are also useful in ...
0
votes
2answers
79 views
What are some applications of smoothing a piecewise polynomial?
What are some applications of smoothing a piecewise polynomial?
For example, I am interested in learning from you:
1) In what future areas of my math studies will this be useful?
and
2) Are there ...
1
vote
1answer
234 views
Discriminant of derivative of cubic equation being a perfect square
Is it possible for the discriminant of the first derivative of a cubic polynomial (x+a)(x+b)(x+c), where a, b and c are distinct non-zero integers (i.e. Discriminant[d((x+a)(x+b)(x+c))/dx] in ...
0
votes
0answers
109 views
How to find similar polynomials to satisfy certain boundary conditions on derivatives
Good afternoon mathematics members. I have one problem with polynomial functions.
For these functions, I need to find similar like in attachment. Derivative in start and end point and boundary ...
-2
votes
1answer
71 views
How to change Hermite function to have a same shape with moving extremum point and zeros
If somebody has a experience with polynomials.
How to set this Hermite function to have a general minimum where I want on $x$ axis, for example in $0.5$. Is it possible to be in analytic form ...
2
votes
2answers
155 views
Prove that the line $y=2x$ intersects the cubic curve $y = x^3 - x + 1$ in at least three different points
Prove that the line $y=2x$ intersects the cubic curve $y = x^3 - x + 1$ in at least three different points
This is a homework question and I don't know where to begin, how would I go about ...
2
votes
1answer
595 views
Homework Help - AP Calculus - Inverse of Polynomial
I know it is a simple problem but I am having trouble. Here is what I have so far:
Let $f(x) = x^5 + 2x^3 + x - 1$
a) Find $f(1)$ and $f'(1)$
I have a) done. $f(1)$ is $3$ and $f'(1)$ is ...
0
votes
0answers
65 views
inverse problem with Orthogonal polynomials
Let $f(x)$ be the function which can be considered the limit $f(x)= \lim_{n \to \infty} C_n P_{n} (x)$, where $P_{n}(x) $ is the $n$-th orthogonal polynomial with respect to a positive even measure ...
1
vote
1answer
89 views
How do I factor this kind of equations?
I was doing some integration by partial fractions exercises and I found this equation:$$\int_{0}^{1}\frac{x^{3}+1}{x^{4}+4x+3},$$ and I don't know how to factor that in order to compute the partial ...
