2
votes
2answers
63 views

Derivatives of trig polynomials do not increase degree?

Let $c = \cos x$ and $s = \sin x$, and consider a trigonometric polynomial $p(x)$ in $c$ and $s$. The degree of $p(x)$ is the maximum of $n+m$ in terms $c^n s^m$. Is it the case that repeated ...
1
vote
4answers
53 views

Finding $p'(0)$ for the polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$

The question goes as follows: Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p'(0)$ is... What I ...
3
votes
2answers
39 views

Solving $4y^4 - 4x^4 + x + y = 0$ (equation system of partial derivates)

I need help solving the following equation system: $$ \frac{\partial}{\partial x} = 8xy + 4y^2 + \frac{y}{x^2 + y^2} = 0 $$ $$ \frac{\partial}{\partial y} = 8xy + 4x^2 - \frac{x}{x^2 + y^2} = 0 $$ ...
2
votes
1answer
32 views

Strict local extremum without $f'$ “changing signs”

Let $f:\mathbb{R}\to \mathbb{R}$. Is it possible that $f$ has the following properties: $f$ is differentiable in a neighborhood of $a\in \mathbb{R}$ $a$ is a strict local minimum There is no ...
0
votes
1answer
39 views

Specify the values of $p$ and $p'$ for a polynomial

Problem 10-26 from Spivak's Calculus, 4th edition: Let $a_1, \dotsc, a_n$ and $b_1, \dotsc, b_n$ be given numbers. If $x_1, \dotsc, x_n$ are distinct numbers, prove that there is a polynomial ...
1
vote
3answers
91 views

Evaluate $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at $t=1$

I need to find a "nice" formula for the evaluation of $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at t=1, where $d_j \in \mathbb{N}$. I have already proved ...
0
votes
1answer
58 views

How to find the roots of the second derivative of $ f(x)=x^2(x − 3)^3$?

I am horrid at factoring and I have to find the inflection points of $ f(x)=x^2(x − 3)^3$. So I to find the inflection points I need to set $f'$ equal to $0$ So I have ...
0
votes
0answers
37 views

How can I cleverly use the error term of polynomial interpolation?

Let $f(x):=x^2$. We're interested in the closed form of the error $|I(f)-T_n(f)|$ where ...
1
vote
1answer
36 views

Formula alteration

is there any way to transform the formula$ \frac {1-x}{x-3}$ into something that can be easily sketched, or which will help eliminate $x$ from the denominator?
2
votes
0answers
112 views

Sign of the derivatives of a simple function

Consider the function $f(x)=x^b(1-x)^{1-b}$ defined on $[0,1]$, with $0 < b <1$. How can we prove that the even derivatives $f^{(2k)}$ have a constant sign on $(0,1)$? One can show that this ...
7
votes
1answer
83 views

Find the maximum value of $ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $

If $x\in\mathbb{R}$ find the maximum value of $$ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $$ I tried this: Let $$y= \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$$ For maxima ...
1
vote
3answers
41 views

How to establish these two facts about polynomials?

Let $f(x) := \sum_{k=0}^n c_k x^k $ be a polynomial of degree $n\geq 0$ with real coefficeints such that $f(x) = 0$ for $n+1$ distinct real values of $x$. Then how to prove that each $c_k = 0$ and ...
0
votes
2answers
45 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
5
votes
1answer
109 views

There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
1
vote
1answer
21 views

Cardinality of a set with a recurrence relation.

Let $A = \left\{ f\in \mathbb{N}\rightarrow \mathbb{C} \mid \forall n\in \mathbb{N}. f(n+3) + 3f(n+1) = f(n+2)+f(n) \right\}$ What is $\left|A\right|$? Well, I tried to treat $f$ as a recurrence ...
0
votes
2answers
43 views

Finding a quadratic equation from roots (Vieta's formula)

Given $x_1 = 1-\sqrt 3$ and $x_2=1+\sqrt 3$, What is the quadratic equation? By Vieta's formula: $-\frac{b}{a} = 1-\sqrt 3 + 1+\sqrt 3 = 2$. Hence, $-b = 2a$ $-\frac{c}{a} = (1-\sqrt 3)(1+\sqrt 3) ...
2
votes
2answers
72 views

$f(x) = x^3 - x$ then $f(n)$ is multiple of 3

If $f(x) = x^3 - x$ then $f(n)$ is multiple of 3 for all integer $n$. First i tried $$f(n) = n^3-n=n(n+1)(n-1)\qquad\forall n\ .$$ When $x$ is an integer then at least one factor on the right is ...
6
votes
2answers
192 views

Find the maximum possible value.

For all ordered triples $(p,q,r)$ define the polynomial $$f_{p,q,r}(x)=x^3-px^2+qx-r$$ Let $a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}$ be (not necessarily distinct) positive reals such ...
0
votes
1answer
55 views

Bounds and uniqueness of a transcendental equation

Let $p\in[0,1]$ and $\rho(x): [0,1] \rightarrow [0,\infty)$ such that $$\int_0^1 dx \rho(x) = 1.$$ I'd like to investigate the following transcendental equation: $$\frac{1}{2p} = \int_0^{1} dx ...
3
votes
1answer
84 views

Proving $\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$

I wish to prove the following inequality for $x\ne 0$: $$\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$$ Using the fact that I already prove: $$\cos x > 1 - \frac{x^2}{2}$$ My try: $\cos x = 1 - ...
0
votes
0answers
49 views

Indefinite integration of general polynomial. Is this correct?

I was reading some notes of a guy I was tutoring the other day on basic calculus. He noted that if $$\int{x^n dx}=\frac{x^{n+1}}{n+1}+c,$$ then that can be extrapolated to all polynomials. He wrote, ...
0
votes
0answers
11 views

Learning a multivariate polynomial with dependent coefficients

I have a polynomial of the form of $ K^2((a-i)^2 + (b-j)^2 + c^2) = (ct)^2$ where $a,b,c,t$ are unknowns. I have multiple observation points for the values of $i,j,K$. Can I use some technique to ...
1
vote
1answer
20 views

relation between the number of real roots of the derivative and the original polynomial

If the derivative of polynomial has n real roots then can we conclude that the original polynomial has to have n+1 real roots?
0
votes
1answer
40 views

conclusion about roots for positive derivative of a polynomial

If the derivative of a polynomial is always positive then what can we conclude about the number of real roots the original polynomial?
1
vote
3answers
72 views

Relation between the roots of $x^2+x+1$ and its derivative

If $f(x)$ is a polynomial in n degree and has $n$ real roots then is it necessary that $f'(x)$ has to have $n-1$ real roots? If this is so then $x^2+x+1$ has no real roots but the derivative of the ...
1
vote
2answers
83 views

Using telescoping property to prove difference of powers

Ok so I have started working through Apostol calculus and as you can see I am stuck. The problem is that I can not see the telescoping pattern anywhere for following problem. Prove that $$a^n - b^n ...
1
vote
3answers
42 views

Inequality for quartic polynomial depending on a parameter

Let $f(x) = \frac 14 x^4 - \frac \alpha2 x^2 - (\alpha-1) x - \frac \alpha 2 + \frac 3 4 $. I want to show that there exists an $\alpha>1$ such that $f(x)\geq 0 $ for $x\leq 0$. Even more, it ...
1
vote
4answers
68 views

Constructing a polynomial with certain zeroes.

I want to construct a polynomial $f(x)$ that has zeroes at $-9,\,-5,\,0,\, 5,\, 9$. Can somebody provide a method (or perhaps some hints) for solving this?
2
votes
1answer
265 views

How to make this polynomial the zero polynomial?(recursively)?

Given a fixed $\beta \in \mathbb{R}$, I want to find the $c_0,...,c_n$ for arbitrary $n \in \mathbb{N}$ such that the polynomial \begin{align}P_n(z):=z(1-z) ...
1
vote
1answer
46 views

Find the Taylor polynomial of degree 4 for cos(x), for x near 0

I am self studying calculus and I need help solving a Taylor Series problem. 1a) Find the Taylor polynomial of degree 4 for cos(x), for x near 0: I think the answer would be: ...
1
vote
1answer
43 views

Question on the norms

I got stuck with the following (simple) question since the result I got seems to be counterintuitive: I have a function defined in terms of its Chebyshev expansion, i.e. ...
0
votes
1answer
29 views

Can a non-linear polynomial have integers values at only integer points?

P(x) be a polynomial with real co-efficient s,deg(P) greater than or equal to 2.Prove that it is not possible that whenever P(x) is an integer,x is also an integer. I tried it in various way and I ...
0
votes
0answers
32 views

Finding roots of a fractional exponential equation.

If we consider a polynomial equation its easy to find the number of roots associated with the expression by applying Descartes Rule. This method, however, doesn't work with non integer exponents. ...
2
votes
4answers
78 views

Limit of a rational function to the power of x

Ok so I have been trying for days already to find a solution to this all around the web and in math books but to no success. The problem is to evaluate a limit of a function composed by polynomial ...
4
votes
2answers
74 views

Is limits $\lim_{x\to\infty}{(x-2)^2\over2x+1}=\dfrac{1}{2}$?

$$\lim_{x\to\infty}{(x-2)^2\over2x+1}=\dfrac{1}{2}$$ I used an online calculator and it said it was actually $=\infty$ Here's how I calculate it: ...
0
votes
1answer
26 views

Use economisation to find linear approximation to x^2-x-1?

I've been given the solution to this question... It uses chebychev, and you get: $1/2(2x^2-1)-2x-1/2$ So the Chebyshev economisation polynomial is $-2T1 -1/2 T0$ I can see the logic in how this ...
4
votes
2answers
142 views

Proof polynomial has only one real root.

I need to prove that this polynomial equation: $$x^5-(3-a)x^4+(3-2a)x^3-ax^2+2ax-a=0\quad\text{ for }\quad a\in(0,\frac{1}{2}).$$ has only one root. That it has one real root is obvious because it is ...
23
votes
1answer
528 views

An awful identity

We take place on $\mathbb C(x_1,...,x_r,x'_1,...,x'_p,u_0,...,u_r,u'_0,...,u'_p)$ with $r,p\in \mathbb N$ Show that : $$\displaystyle{\sum_{i=1}^r \left( \frac{\prod_{j=0}^r (u_j-x_i) ...
1
vote
2answers
53 views

Describing asymptotic behaviour of a function

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution
5
votes
1answer
96 views

Analyzing a fourth degree polynomial

Let $a,b$ and $c$ be real numbers. Then prove that the fourth degree polynomial in $x$ $acx^4+b(a+c)x^3+(a^2+b^2+c^2)x^2+b(a+c)x+ac$ has either 4 real roots or 4 complex roots. I have never solved a ...
7
votes
2answers
303 views

Real roots of a polynomial

Let $p$ be an even degree polynomial with real coefficients such that the product of the constant term and the leading coefficient is negative. Show that $p$ has at least two real roots. Thanks!
1
vote
1answer
69 views

A Cubic Equation

$2x^3+ax^2+bx+4=0$, $(a,b \in R^+)$ has three real roots. Then : A. $a\geqslant 4.2^{\frac 1 3}$ B. $a\geqslant 1.2^{\frac 1 3}$ C. $a\geqslant 6.2^{\frac 1 3}$ D. $a\geqslant 2.2^{\frac 1 3}$ ...
0
votes
1answer
45 views

Bounds on coefficients of close polynomials

I've got two polynomials $p, \hat{p}:\mathbb{R}^2\rightarrow \mathbb{R}$ of degree $2\times2\ $ which are close together around $0$: $$|p(\mathbf{x})-\hat{p}(\mathbf{x})|<\varepsilon \quad \forall ...
1
vote
0answers
140 views

Solving an 8th degree polynomial

I know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...
6
votes
1answer
82 views

Show that $P$ is divided to simple roots knowing that $a_{k}^2-4a_{k-1}a_{k+1}>0$

Let $P(X)=a_0+a_1X+..+a_nX^n\in R[X]$ Assume that $\forall k, a_k>0$ and $a_{k}^2-4a_{k-1}a_{k+1}>0$ Show that $P$ is divided to simple roots in $R[X]$. i.e. ...
0
votes
1answer
40 views

Inferring a characteristic of a ratio of functions from the ratio of their derivatives

This is a strange one, but I need help trying to understand whether there is any logic behind this or not. Given $\frac {f(\sqrt{2})}{g(\sqrt{2})}=2$, and $\frac {f'(x)}{g'(x)}>2$ for all ...
14
votes
6answers
465 views

Proving $x^{4}+x^{3}+x^{2}+x+1$ is always positive for real $x$

So I was bored in class and decided to graph polynomials in geogebra, I noticed that $x^{4}+x^{3}+x^{2}+x+1$ and $x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$, are all above the x-axis. Now I am wondering if ...
0
votes
1answer
39 views

Leading coefficient of a polynomial quotient is equal

Isn't there something special that you can infer when a polynomial's leading coefficients are equal on the numerator and denominator, like this: $\frac{x^4}{x^4+3x^4}$ I'm trying to find the limit ...
0
votes
1answer
40 views

How to simplify floor polynomial given lower bound on x?

$$ \left\lfloor\frac{8x^2 + 5x -4}{3x^2 + x}\right\rfloor $$ where $x$ > $\sqrt{8}$ How would you simplify this type of expression? *Please note the floor operation surrounding the expression ...
0
votes
3answers
59 views

How to generically solve polynomial expressions given a minimum value of x?

Given something like $\dfrac{8x^2 + 2x + 7}{3x^2 + 2x}$, and $x > \sqrt8$, what strategy would you employ to simplify this expression?