Root in (0,1) (0,1)

Let a0,a1,,an a_0, a_1, \ldots, a_n be real numbers which satisfy

ann+1+an1n++a23+a12+a0=0. \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

anxn+an1xn1++a1x+a0=0 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) (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.

Note by Calvin Lin
7 years ago

No vote yet
1 vote

  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.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. 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]( 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"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

Consider the polynomial function : f(x)=anxn+1n+1+an1xnn++a1x22+a0x.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)=0f(0)=0, and f(1)=0f(1)=0 by the first given equality. By Rolle's theorem there is some x(0,1)x\in (0,1), such that f(x)=0f'(x)=0 which can be written as : anxn+an1xn1+a1x+a0=0.a_n x^n +a_{n-1}x^{n-1}+\dots a_1 x+a_0=0.

Haroun Meghaichi - 7 years ago

Log in to reply

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 ago

Log in to reply

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 ago

Log in to reply

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

Vishal Sharma - 7 years ago

Log in to reply


U Z - 6 years, 2 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...