Root in $ (0,1) $

Let $a_0, a_1, \ldots, a_n$ be real numbers which satisfy

$\frac{a_n} { n+1} + \frac{a_{n-1} } {n} + \ldots + \frac{a_2}{3} + \frac{a_1} { 2} + a_0 = 0 .$

Show that the equation

$a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 = 0$

has at least one solution in the interval $(0,1)$ .

This is a list of Calculus proof based problems that I like. Please avoid posting complete solutions, so that others can work on it.

#Calculus #Proofs

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Consider the polynomial function : $f(x)=\frac{a_n x^{n+1}}{n+1}+\frac{a_{n-1}x^n}{n} +\dots +\frac{a_1x^2}{2}+a_0 x.$ Clearly $f(0)=0$ , and $f(1)=0$ by the first given equality. By Rolle's theorem there is some $x\in (0,1)$ , such that $f'(x)=0$ which can be written as : $a_n x^n +a_{n-1}x^{n-1}+\dots a_1 x+a_0=0.$

Haroun Meghaichi - 7 years, 2 months ago

Good work! I did the same, but with due respect "please avoid posting complete solutions, so that others can work on it"

Piyal De - 7 years, 2 months ago

It appears to b a problem based on ROLLE's THEOREM........ On integrating the given equation we get a new equation whose roots are 0 &1. Now the new equation is a polynomial so it is continuos as well as differentiable in (0,1) . Now apply ROLLE'S THEOREM which states that between any 2 roots of a continuous and differentiable function there must b atleast one root of its differentiable function.

Abhishek Pal - 7 years, 2 months ago

Rolle's theorem for interval (0,1). :)

Vishal Sharma - 7 years, 2 months ago

JEE PROBLEM

U Z - 6 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Root in \( (0,1) \)

Comments