# Questions tagged [rolles-theorem]

For questions about Rolle's Theorem, or exercises that suggest the use of Rolle's Theorem. The theorem states that if a real-valued differentiable function has two distinct zeros, then the function has a vanishing derivative for some value between those two zeros.

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### Show that the polynomial $f(x) = 3x^4 - 2 x^3 + x^2 + ax - 1$ with $a \in \mathbb{R}$ does not have all of its roots real.

Consider the polynomial: $$f(x) = 3x^4 - 2x^3 + x^2 + ax - 1$$ with $a \in \mathbb{R}$ and the roots $x_1, x_2, x_3, x_4 \in \mathbb{C}$. I have to show that the polynomial $f$ cannot have all of ...
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### Find the minimum number of zeros of $g(x)=(f'(x))^2+f(x)f''(x)$in the interval $[a, e]$ for the following given information.

If $f(x)$ is twice differentiable function such that $f(a)=0, f(b)=2,f(c)=–1, f(d) =2,f(e)=0$, where $a<b<c<d<e$, then find the minimum number of zeros of $g(x)=(f'(x))^2+f(x)f''(x)$in ...
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### Mean Value Theorem demonstration

$\large f(x)=\tan^{-1}(\frac{1}{x^2}) -\ln(x^2+1)$ $\large if 1\leq x < y$ $\large \text { and } 1-\frac{2x}{1+x^2}\geq 0$ Demonstrate that $\large |f(x)-f(y)|\leq 2|x-y|$ I managed to ...
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### Prove that the second derivative has a zero

my question is: Can I use Rolle's theorem to prove that the second derivative has a zero? Consider a Real function of Real variable defined by $f(x)=(x+1)\cdot e^{x^2}$. Prove that the second ...
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### Using Rolle's theorem prove that if $b^2<3ac$, then there is exactly one root to $f(x) \equiv ax^3+bx^2+cx+d=0$

Using Rolle's theorem prove that if $b^2<3ac$, then there is exactly one root to $f(x) \equiv ax^3+bx^2+cx+d=0$ I literally have no idea how to use Rolle's Theorem. Rolle's Theorem: if the ...
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### Rolle's Theorem 1

I need your help. I don't know how to translate exactly the math problem I have been given for homework, since English is not my mother language, so I would really appreciate it if you didn't judge me....
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### What is the maximum possible value for $f(x)$ for $x \in [0,1]$? [closed]

A function $f(x)$ is continuous and differentiable in $[0,1]$. If $f'(x) \le 10$ for all $x \in [0, 1]$ and $f(0) = 0$, what is the maximum possible value of $f(x)$ for $x$ in $[0, 1]$?
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### Trying to apply Rolle's Theorem

Let $f$ be a differentiable function in the interval $[a,b]$. Prove that there exists $c\in ]a,b[$ such that $$f'(c)=f(c)\dfrac{(a+b-2c)}{(c-a)(c-b)}$$ My ideas are: maybe we need to find a ...
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### Rolle's theorem for second derivative

A problem asks the following $f$ is a twice-differentiable function on some segment $[a,b]$ such that $f(a)=f(b)$ and $f'(a)f'(b)<0$. it asks to prove that the second derivative of $f$ vanishes ...
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### Let a function f(x) be twice differentiable such that $f(0) = 0, f(\pi/2)= 1 , f(3\pi/2)=-1$. To prove that there exists a ‘c’ in $(0,3\pi/2)$

Let a function f(x) be twice differentiable such that $f(0) = 0, f(\pi/2)= 1 , f(3\pi/2)=-1$. To prove that there exists a ‘c’ in $(0,3{\pi/2})$ such that |$f”(x)$ | is less than or equal to 1. ...
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### Is this inference from Rolle's Theorem correct?

In a script I'm reading right now that is providing a proof for Cauchy's mean value theorem (extended mean value theorem), it says "We have $g(x) \neq 0$ for all $x\in ]a,b[$ (otherwise, there would ...
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### Application of Rolle's theorem and determinant

$f(x) = \begin{vmatrix} \sin^3x & \sin^3 a & \sin^3 b \\ xe^x & ae^a & be^b\\ \frac{x}{1+x^2} & \frac{a}{1+a^2} & \frac{b}{1+b^2} \end{vmatrix}$ Where $0<a<b<2π$ ...
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### Rolle's theorem proof in Apostol: meaningfulness of interior

According to the statement of the Rolle's theorem in Apostol calculus 1, we need to have a continuous function on $S = [a, b]$, and this function should have a derivative on the interior of $S$. I do ...
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### Some property of differentiable function

Let $f:[0,2]\to\mathbb{R}$ be a continuous function and $f$ is differentiable on $(0,2)$, and let $f(0)=f(2)=0$. Now, suppose that there is a point $c\in(0,2)$ such that $f(c)=1$. Then, there is a ...
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### Spivak, Calculus 3rd Ed. Chapter 11 problem 32

I made it this far through the book but now I'm really stumped. Here's the problem: Suppose that $f$ and $g$ are two differentiable functions which satisfy $fg'-f'g=0$. Prove that if $a$ and $b$ ...