# Tagged Questions

44 views

### How do I go about factoring this polynomial?

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 ...
33 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 ...
35 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?
92 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 ...
75 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 ...
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 ...
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 ...
108 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 ...
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 ...
37 views

61 views

### Spivak Calculus problem with inductive proof of polynomial property

Ok so I have been trying to do this proof of what seems to be a version of Factor theorem(I am now well versed in math so please forgive me if I am wrong). Problem goes like this: Prove that ...
80 views

### $p(x)\geq 0 \forall x\Rightarrow p(x)+p'(x)+p''(x)+…+p^{(n)}(x)\geq 0$ [duplicate]

$p(x)\geq 0 \forall x \in \mathbb{R} \Rightarrow p(x)+p'(x)+p''(x)+...+p^{(n)}(x)\geq 0$, where p(x) is a polynomial of degree n. I showed: $a_{n}+...+a_{0}\geq 0$, ...
96 views

### If something holds for n+1 distinct values then it holds for all values.Proving a property of polynomial

Ok I am stuck on proving a property of polynomial.It basically goes like this. If $$f(x) = \sum_{k=0}^{n} c_kx^k$$ is equal to zero for n+1 distinct real values x,then f(x) is equal to 0 for all ...
### Solve $t^4+4 t^3+6 t^2+4 t-32 t^{1/4}+1 = -16$
I'm trying to solve the following equation: $$(t+1)^4 - 32 t^{\frac{1}{4}}=-16$$ where t $\geq 0$, which is equivalent to $$t^4+4 t^3+6 t^2+4 t-32 t^{\frac{1}{4}}+1 = -16$$ Wolfram Alpha tells that ...