# Tagged Questions

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 ...
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 ...
39 views

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

I need help solving the following equation system: $$\frac{\partial}{\partial x} = 8xy + 4y^2 + \frac{y}{x^2 + y^2} = 0$$ $$\frac{\partial}{\partial y} = 8xy + 4x^2 - \frac{x}{x^2 + y^2} = 0$$ ...
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 ...
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 ...
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### 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 ...
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### 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 ...
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 ...
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?
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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...