2
votes
1answer
25 views

Solving $\operatorname{ctg} x=x/b$

I have no problems finding first solution (both: $b \to 0$ and $b \to \infty$). My solutions on photos. I got stuck trying to find solution when $x \to \infty$. As I think, solution for $x$ will have ...
0
votes
0answers
27 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. ...
1
vote
3answers
101 views

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 ...
1
vote
1answer
36 views

How do I find the roots of a quartic, without guessing?

I'm given a quartic function to sketch, and one of the things to find is the zeros/x-roots/x-intercepts. After a lot of guessing (and no success) I pulled it up on my trusty TI, to find the roots are ...
5
votes
1answer
77 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 ...
1
vote
0answers
46 views

Number of zeros of Wronskian

Is there some relation between the number of zeros of a Wronskian and properties of given functions? Having Wronskian (e.g. $2$ x $2$) $$W(x)=\left|\begin{array}{c}f_1(x) & f_2(x)\\f'_1(x) & ...
1
vote
0answers
81 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 ...
5
votes
1answer
181 views

Limits of the solutions to $x\sin x = 1$

Let $x_n$ be the sequence of increasing solutions to $x\sin{x} = 1$. Define $$a = \lim_{n \to \infty} n(x_{2n+1} - 2\pi n) $$ and $$b = \lim_{n \to \infty} n^3 \left( x_{2n+1} - 2\pi n - \frac{a}{n} ...
2
votes
4answers
92 views

Rolle's theorem prove polynomial has only 1 root

Prove that $x^3-x-4=0$ has exactly one real root: This is my working so far: suppose $f(x) = x^3-x-4$ has $2$ roots : $a,b$ $f(a) = f(b) = 0$ $f'(x)=3x^2-1$ $f'(x)$ exists on $(a,b)$ so $f$ is ...
2
votes
1answer
29 views

Extend rolle's theorem to complex functions?

If $f(z)$ is a polynomial of degree n with n distinct real roots $r _1$<,..., <$r_n$, then there exists exactly one root of $f '(z)$ in between any consecutive root of $f(z)$. The context of ...
0
votes
1answer
79 views

Finding the roots of the derivative of a trig function

I have a function of which i need to find the maximum of. The function is : D(t)= 12.17- 1.15cos(2*pi*t/365) + .18 sin(2*pi*t/365) Taking the derivative with respect to t, I correctly get: D'(t) = ...
2
votes
3answers
141 views

If $x\in\mathbb R$, solve $4x^2-40\lfloor x\rfloor+51=0$.

If $x\in\mathbb R$, solve $$4x^2-40\lfloor x\rfloor+51=0$$ where $\lfloor x\rfloor$ denotes the integer part of the number. $\lfloor x\rfloor\le x$ and $\lfloor x\rfloor=x-\{x\}$, where $\{x\}$ ...
4
votes
2answers
58 views

Newton iteration method

i need some help here. My function is $f(x) =x^{3}$ . I was asked to find the number of iterations that are needed to reach the precission $10^{-5}$ if $x_{0} = 0.9$ I was wondering if there is a ...
2
votes
2answers
72 views

Question about bisection method

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 ...
1
vote
2answers
36 views

Lemma 2.5.5 Boas, Entire functions

I'm reading Boas, Entire functions, but I don't understand lemma 2.5.5, which states that $\sum_{1}^{+\infty}\frac{1}{r_{n}^{\alpha}}$ and the integral $\int_{0}^{+\infty}t^{-\alpha -1}n(t)dt$ ...
0
votes
1answer
80 views

Find the minumum using Newton-Raphson

I have the following function: $f(x) = 100(x_2 - x_1^2)^2 + (1-x_1)^2$ I have to find the minimum of this function using the Newton Raphson method. The point where I have to start is $x = [1.2$, ...
17
votes
6answers
2k views

Can $x^3+3x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem: Solve $x^3+3x^2+1=0$ Am I missing something here, or is indeed a more ...
6
votes
0answers
56 views

If all convex combinations of $p(x)$ and $q(x)$ have real roots, then $p,q$ have a common interlacing poly

I heard this result in a talk the other day: Suppose $p$ and $q$ are polynomials. Suppose $p$ is a polynomial of degree $n$ and $q$ a polynomial of degree $n-1$. Call $q$ and interlacer of $p$ if the ...
3
votes
3answers
165 views

How find this equation solution $2\sqrt[3]{2y-1}=y^3+1$

find this equation roots: $$2\sqrt[3]{2y-1}=y^3+1$$ My try: since $$8(2y-1)=(y^3+1)^3=y^9+1+3y^3(y^3+1)$$ then $$y^9+3y^6+3y^3-16y+9=0$$ Then I can't.Thank you someone can take hand find the ...
1
vote
3answers
560 views

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 ...
0
votes
0answers
34 views

Formula for roots on an arbitrary polynomial

This question is related to a previous question I have posted: Solution for a Mixture of Two Exponential Equations I have reduced my problem to the following equation: $\left( ...
3
votes
1answer
130 views

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 ...
0
votes
0answers
30 views

Convergence of $xe^x - R$

Basing my question on one of the previous questions I have passed before Root of the function $f(x)=xe^x-R$, I was wondering why does $xe^x - R$ always converge? I was told that the function will ...
0
votes
1answer
75 views

Finding root between two points

The function $f:[0,1]\to \mathbb{R}$ is continuous, $f(0)<0$, $f(1)>0$ and there is one root in between. Using $f(0)$ and $f(1)$, the expression $\frac{1\cdot f(0)-0\cdot f(1)}{f(0)-f(1)}$ would ...
0
votes
3answers
211 views

$\log x =Cx^4$ has only one root. Find C

$\log x =Cx^4$ has only one root. Find C. I don't know how to solve this problem. Do you take derivative on both sides? I am thinking C equals 0. Am I correct on that?
11
votes
1answer
206 views

Solving $x^2+bx^{1+\varepsilon}+c =0$

Let $x \in \mathbb{R}$. Is it possible to find the roots of $x^2+bx^{1+\varepsilon}+c =0$ where $b,c \in \mathbb{R}$ and $\varepsilon$ is small. I am guessing that an explicit expression might not be ...
3
votes
2answers
111 views

Root of the function $f(x)=xe^x-R$

How can we find the root of the function $f(x)=xe^x - R$ for a general R where $R>=-1/e.$ I don't have any idea as to how to even approach this. Came across this problem during my self-study in ...
1
vote
2answers
112 views

verifying a polynomial is positive on the half-line

Math people: I am running experiments that produce polynomials $P(z)$ that, in every experiment I have run, are always positive on the half-line $\{z \geq 1\}$. I want to prove analytically that the ...
4
votes
2answers
95 views

What this sine function equation means?

Apostol's book "Calculus" asks to prove that $$\sin\frac{\pi }{6}=\frac{1}{2}$$ using the fact that $$\sin 3x=3\sin x-4\sin^3 x$$ and $$\sin \frac{\pi}{2}=1$$ So, we take $x=\frac{\pi}{6}$ and ...
4
votes
1answer
163 views

Solutions of the equation $\tan x=x$ over $\mathbb{R}$

I know it can be solved by computation but I was looking for a closed form solution. For me computation means we don't know and sooner or later we will find closed form solutions to every problem or ...
5
votes
0answers
108 views

Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $f_1(x)=x,f_2(x)=x^x,\dots f_{n+1}(x)=x^{f_n(x)}~(n\geq 1).$ Let $F_n(x)=f_n^{'}(x).$ Hence $F_1(x)=1, F_2(x)=x^x(1+\log(x))\dots.$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
0
votes
1answer
152 views

The function defined by $f(x)=\sin\pi x$ has zeros at every integer. Show that when $-1<a<0$ and $2<b<3$ , the bisection method converges to

The function defined by $f(x)=\sin\pi x$ has zeros at every integer. Show that when $-1<a<0$ and $2<b<3$ , the bisection method converges to (a) $0$, if $a+b<2$ (b) $2$, if $a+b>2$ ...
0
votes
1answer
75 views

Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires that $n>1000$.

Let $f(x)=(x-1)^{10}$. The root of the equation , $p=1$. The approximates of the root, $p_n=1+\frac{1}{n}$ Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires ...
0
votes
1answer
69 views

Difficulty to solve the exercise of Bisection method.

Find an approximation to $ {25}^{\frac{1}{3}}$ correct to within $10^{-4}$ using the Bisection algorithm. How to solve it? Where are the function and interval here?
0
votes
3answers
276 views

I am not understanding what has asked to compute of the following exercise.

let $f(x)=(x+2)(x+1)x(x-1)^3(x-2)$. To which zero of $f$ does the Bisection method converges when applied on the interval $[-3,2.5]$ Have i asked to find the root of $f(x)$ ?
0
votes
1answer
57 views

Determine the number of iteration to find solutions accurate to within $10^{-2}$ for $f(x)=x^3-7x^2+14x-6=0$ on $[a,b]=[1,3.2]$

i got the number of iteration,$n$, to achieve the accuracy, $\epsilon=10^{-2}$ is $n=5.5\approx 6$ But in answer script, $n=8$. My procedure is $ \frac{(b-a)}{2^n}<\epsilon$ ...
2
votes
1answer
94 views

Correct answer of the following math related to Bisection Method.

Use the Bisection method to find $p_3$ for $$f(x)=\sqrt x-\cos(x)$$ on $[0,1]$ I have got the answer $p_3=0.875$ But in answer script , $p_3=0.625$ Which one is correct? let $[a,b]=[0,1]$ ...
0
votes
1answer
71 views

Roots of $x^{2}+e^{0.1x}-1$

I saw an exercise that asks to prove that $f(x):=x^{2}+e^{0.1x}-1$ have a root $r<0$. The solution stated that $f''(x)=2+(0.1)^{2}e^{0.1x}>0$ hence there is a maximum of two roots, since $0$ is ...
2
votes
3answers
989 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!
4
votes
2answers
113 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??
3
votes
2answers
205 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 ...
1
vote
2answers
73 views

Indeterminate Limit (Finding the Remainder to a Root)

So i was working on this: $$ \lim\limits_{x\to1} \frac{x + \sqrt{x} - 2}{x - 1} $$ and I thought to simpify my top by multiplying by a conjugate, taking everything other than the $x$ to be the $b$ ...
1
vote
1answer
53 views

Root of sum of shifted polynomials

For an arbitrary positive odd integer $k$, I would like to obtain an expression for the root $x_{root} \in \mathbb{R}$ of the following polynomial $$p(x) = \sum_{i=1}^N (x-x_i)^k,$$ where $x_i\in ...
3
votes
2answers
123 views

How to show that there exists a root of $f(x)=0$?

Let $f(x)=\sum_{k=0}^n a_k x^k$, where $a_k$'s satisfy $\sum_{k=0}^n \frac{a_k}{k+1}=0$.Show that there exists a root of $f(x)=0$ in the interval $(0,1)$.
0
votes
3answers
194 views

number of roots of an equation

Plotting the equation $x^3-x^2 \sin(x)+\cos(x)$ I see that $x^3-x^2 \sin(x)+\cos(x)=0$ has only one real solution, is there a simpler way to see that it cannot have 3 real solutions?
0
votes
2answers
294 views

How to do this Intermediate value theorem proof?

Use the Intermediate Value Theorem to show that the equation $x^3+x+1=0$ has a solution. How to do this? :S Thank you very much!
0
votes
2answers
50 views

roots written as exponents

It appears that i'm not quite sure anymore how to write roots as exponents, and how to work with them. I know $\sqrt[3]{a}$ is written $a^{\frac{1}{3}}$, but I don't know how to handle them in things ...
4
votes
1answer
2k views

How do you prove an equation has one root?

I have this equation: $$9x + \cos x = 0$$ but I need to write out and prove why it has one real root. Could someone maybe give me a few pointers or what do I do exactly?
24
votes
2answers
635 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 ...
10
votes
7answers
3k views

How to solve an exponential equation with two different bases: $3^x - 2^x = 5$

Can anyone tell me how to solve this equation $$3^x - 2^x = 5$$ other than graphically? I'm stunned. I don't know what to do in the first step.