# Tagged Questions

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### Existence of a root

Let $f:[a,b] \rightarrow \Bbb R$ continuous, such that for every $x$ there is a $y$ such as that $|f(y)|\leq|f(x)|/2$. Show there exists a $\xi$ such that $f(\xi)=0$
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### Understanding Multiplicities

I am having troubles understanding what 'multiplicities' mean. In example what does $-1/3(multiplicity 2)$ translate into?? To clarify this is for finding zero's in a polynomial function Any help ...
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### Function such that zeros$=$order of the derivative

Does there exist a function $f\in C^n(\mathbb{R},\mathbb{R})$ for $n\ge2$ such that $f^{(n)}$ has exactly $n$ zeros, $f^{(n-1)}$ has exactly $n-1$ zeros and so on ? Where $f^{(n)}$ is the nth ...
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### Roots of $x \cos{x}-1$ and $\cos{x}-x^{-1}$

To finding the roots of $x \cos{x}-1=0$ we can write the equation as $$f(x)=x \cos{x}-1=0 \to x \cos{x}=1 \to \cos{x}=\frac{1}{x} \to \cos{x}-\frac{1}{x}=u(x)-v(x)=g(x)=0$$ The roots of $f(x)$ are ...
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### roots of sum of exponential functions

Could anyone point me in the right direction of finding the roots of equations of the form $$\sum_{i=1}^n a_ie^{f_i(x)},$$ where $a_i \in \mathbb{R}$ and the $f_i$ are each first degree polynomials ...
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### Roots of real polynomial

$f(x)$ is a real polynomial. Show that $z=a+bi$ and $\bar z=a-bi$ have the same algebric multiplicity. I know that if $z=a+bi$ is a root of $f$ then $\bar z=a-bi$ is too, but don't know how to use ...
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### The function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$.

Show that if $f\in \mathcal{C}^3$ and $2\cdot\pi$ periodic then the function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$. My attempt : f is $2\pi$ periodic and $\mathcal{C}^3$, we have : ...
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### Estimate error when using the result of an expanded equation

I would like to know how to deal with the error term in expanded expression. For example consider the function $\displaystyle f(x)=A\text e^{-(x+\lambda)^2}+B\text e^{-(x-2\lambda)^2}\;,$ where ...
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### Find roots of a function

$f$ is a function defined on the whole real line which has the property that $f(1+x)=f(2-x)$ for all $x$. Assume that the equation $f(x)=0$ has $8$ distinct real roots. Find the sum of these roots. I ...
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### Find amount of roots

We are given equation $$\frac {e^x}{x^2} = a$$. Task is to find how many solutions equation would have depending on values of a. Let's illustrate a(x): It's easy to conclude that there aren't no ...
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### Effect on roots of function on taking the derivative of the function

Suppose there is a function $$f(x)=(x-1)^{15}(x-2)^{20}(x-3)^{25}(x-4)^{30}$$ As we take the derivatives of the function, what will happen to the number of real roots and the number of distinct real ...
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### Determine the number of zeros in the first quadrant

This is a homework question: $$f(z) = z^2 - z + 1$$ sorry for the poor code!
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### Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.

Prove that the function $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$. My work so far: $f(0)=0$ Thus, $x=0$ is a root. For the ...
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### Padé approximant of transfer function with gain and time delay.

$$H(\omega) = A e^{-j \omega \tau}$$ I'm trying to use Padé approximation to generate a numerator and denominator polynomial for the above transfer function but genuinely struggling with how to ...
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### Let $f(x)=x^2+17x+a$, $g(x)=x^2-17x-a$, $r$ a root of $f$ and $-r$ a root of $g$. Determine the roots of $f$.

Let $f(x)=x^2+17x+a$ and $g(x)=x^2-17x-a$. Suppose $r$ is a root of $f$ and $-r$ is a root of $g$. Determine all roots of $f$. From the descriptions, I can conclude that $f(x)-g(x)=2a$. But that ...
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### Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
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### Finding the Zeros of A Function

In my Algebra II class we are learning how to find the zeros of a function, but I find the process very confusing despite the many efforts of my algebra teacher to explain them to me. I understand ...
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### Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}$\ $(-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$ My attempt : ...
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We have $f(x)=(x-1)^3(x-2)(x-3)$. $a_0<1,b_0>3$. We had to show that if $\frac{a_0+b_0}{2}\ne 1,2,3$, there is one root of $f$ that we can't get it by the bisection method. I guess that this is ...
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### Demostrating that a trascendental function can be expressed as a product of its roots [duplicate]

I'm doing a work about the product of Wallis, a formula to calculate pi that is π/2= ∏ (2n/2n-1)· (2n/2n+1) from n=1 to n=∞. I have to prove the formula, and I have been searching in books and all ...
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### Use the intermediate value theorem to show a function has a root [closed]

Let $f$ be a function defined on $(-\infty, 0)$ by $$f(x) = x^3 + \frac{4}{x^2} + 7 \ .$$ Use the Intermediate Value Theorem to show that the given function has at least one zero in the ...
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### Finding roots of a function in an interval

Does the equation $x^3-12x+2=0$ have three solutions in the interval $[-4,4]$? We know that this is a continuous function because it's a polynomial, and so we can use the Intermediate Value ...
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### Conversion of roots of a polynomial

I'm wondering, given a polynomial $P(x)$ with roots $r_i (1\le i\le n)$, how to determine the polynomial $Q(x)$ such that its roots are $r'_i=f(r_i)$. For example, if $P(x)=x^2-x-6=(x-3)(x+2)$ and ...
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### At which parameter value $c>0$ do the number of solutions of $\log(1+x^2)=x^c$ change?

I'm looking at the functions $x\mapsto \log(1+x^2)$ and $x\mapsto x^c,\ c>0$ on the interval $\mathbb R^+_0$. I'm interested in the properties of $$\log(1+x^2)=x^c.$$ Graphically, for small $c$, ...
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### What are the methods to find approximatly the 5th roots of an equations

By which method, can I find the nearest root of : $x^5−2x+1.1=0$ ? Thank you.
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### Help with finding the roots of a function

I have the following problem: Find the smallest root of the function $e^{-x} = \sin (x)$ and focus the root with Newton's method to $8$ decimal accuracy. Any suggestions? :) Thank you for any ...
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### Solve $x+y+z = x^3 + y^3 + z^3 = 8$ in $\mathbb{Z}$

Solve $x+y+z = x^3 + y^3 + z^3 = 8$ in $\mathbb{Z}$ First I tried to transform this equation, substituting $x = 8-y-z$. So I end up with: $$x^3 + y^3 + z^3 = 8$$ $$(8-y-z)^3 + y^3 + z^3 = 8$$ ...
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### Prove that $f$ has finite number of roots

Let $f:[0,1]\to \mathbb{R}$ be a differentiable function. If there do not exist any $x\in[0,1]$ such that $f(x)=f'(x)=0$, prove that $f$ has only finite number of zeros in $[0,1]$. I'm not ...
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### Finding a function with properties

I am looking for a function $f(x)$ with the following properties: Positive for $x\in(-\infty, 0)$ but tangent to the x-axis at $x=-1$ A root at $x=0$ and negative for $x\in(0, 2)$ A root at $x=2$ ...
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### How to show that a root of the equation $x (x+1)(x+2) … (x+2009) = c$ can have multiplicity at most 2?

How to show that a root of the equation $$x (x+1)(x+2) ....... (x+2009) = c$$ can have multiplicity at most 2 , and to find the value of $c$ for which this is possible. I proceeded by using the ...
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### find number of solution for given equation

suppose we we have following equations and conditions Let $k$ be the number of real solutions of the equation $e^x+x-2=0$ in the interval $[0, 1]$ and and let $n$ be the number of real solutions ...
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### Formula to scale a series that is being bent with a root / power.

I have a reference number, Rx, and a series of numbers, Sx[], to compare to it. Let's call the output Ox[]. I am using a simple square root to accelerate the apparent difference between the reference ...
### Number of roots $|f(|x|)|$ has according to $f$
If $f$ has one root on $(-\infty,0)$ and two distinct roots on $(0,+\infty)$ and $f(0)=-1$, how many roots does $|f(|x|)|$ have? I know graph of $|f(|x|)|$ should be in quadrant I because $x$ ...