# 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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### On Rolle's Theorem

Assuming that the function $f$ is differentiable in $(0,1)$ and continuous on $[0,1]$. If $f(1) = 0$, show that there exists one $c \in (0,1)$, such that $$f(c) = \frac{c f'(c)}{100}.$$ My Attempt: I ...
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### discovering Mean Value Theorem

mean value theorem for single variable function is very easy and intuitive once you "see" the formula. Actually, My question, slightly weird but helpful, is that How does someone come up ...
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### Prove that there exists $\xi \in (a,b)$ such that $f(a)-2f(\frac{a+b}{2})+f(b)=\frac{1}{4}(b-a)^2f''(\xi) .$ [duplicate]

Given that f is twice differentiable on $[a,b]$, prove that there exists $\xi \in (a,b)$ such that $$f(a)-2f\left(\frac{a+b}{2}\right)+f(b)=\frac{1}{4}(b-a)^2f''(\xi) .$$ This problem was given in a ...
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### If for $x\in(\frac{1}{2},\infty)$ we have $f'(x)=(e^x-1)(x-2)(x-3)$. Show that there exist exactly two roots of $f''(x)=0$ in the given domain

Let $f:\left(\frac{1}{2},\infty \right)\to \mathbb{R}$ be a function such that $f'(x)=(e^x-1)(x-2)(x-3)$. Show that there exist exactly two roots of $f''(x)=0$ in the given domain. My Attempt I ...
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### Rolles theorem: differentiabilty open brackets significance

https://math.stackexchange.com/a/2863979/922054 Related to Rolle's theorem : Here it is said that $f(x)$ is non-differentiable at endpoints of $\sqrt{1-x^2}$, but isn't it okay to talk of endpoint ...
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### If $f(x)\rightarrow L$ from both sides then $f'(c)=0$ for some $c$ [duplicate]

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f$ is differentiable over $\mathbb{R}$. Prove that if $\underset{x\rightarrow\infty}{\lim}f(x)=L$ and $\underset{x\rightarrow-\infty}{\lim}f(x)=L$ ...
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### Finding the number of real roots of a polynomial

I want to find the number of real roots of $f(t)=t^4-2t^2+4t+1$. As $f(0)=1>0, f(-1)=-4 <0$ and $f(-2)=1>0$, I can say that there are two real roots since the polynomial is continuous. For ...
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### Differential calculus using Rolles theorem and mean value theorem

Q- A function f is such that its second derivative is continuous on [a, a+h] and Derivable on (a,a+h). show that there exists a number $\theta$ between 0 and 1 such that-  f(a+h) - f(a) -\frac{h[f'(...
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### How do I determine further solutions of the equation using Rolle's theorem?

I gave this equation $2^x=1+x^2$ with the $1$st zero is $x_1=0$ and the $2$nd zero is $x_2=1$. (easy reading) Now I want to calculate more zeros using Rolle's theorem, and I've rearranged the function ...
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### Sum of roots for a even function and slope being 0?

High school senior and was helping my friend with Rolle's theorem and came across this neat point. Given $f(x)$ is a continuous and differentiable function, and $f(-x)=f(x)$. Additionally, $f(x)=0$ ...
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### Rolle's Theorem and error bound formula for trapezoidal rule

In these notes on error bounds for numerical integration, it shows proofs for the error bound formula for the midpoint rule and the trapezoidal rule. I don't understand the last part (page 5-6) for ...
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### Derivative at end point of a interval

While studying Rolle's Theorem, a question came in my mind that can there exist a function which is continuous in the interval [a,b] and differentiable in the interval (a,b) but not differentiable at ...
### Let $f(x)$ be non-constant, twice differentiable function define on $\mathbb R$ such that $y=f(x)$ is symmetric about line $x=1$
Let $f(x)$ be non-constant, twice differentiable function define on $\mathbb R$ such that $y=f(x)$ is symmetric about line $x=1$ and $f(-1)=f'(\frac{1}{4})=f'(\frac{1}{2})=0$ then which of the ...