# 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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### How to do this question with the help of Rolle's theorm [duplicate]

If $a_1$, $a_2$, $a_3$,$\cdots$ $a_n$ $(n\ge2)$ are real and $(n-1){a_1}^2-2na_2<0$ then prove that at least two roots of the equation $$f(x)=x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_n=0$$ are imaginary. ... 18 views

### Integral and number of solutions

I would need some help with these 2 exercises: sin(x)*sin(2x)=1 and x is from 0 to 4pi. Number of solutions for this ecuation. 2 integral from 0 to pi/2 from 1/(sin(x + pi/6)*sin(x+pi/3))
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### a little problem about Rolle theorem

As Rolle theorem goes,if $f(x)$ is continuous and well-defined in $[a,b]$, derivable in $(a,b)$, and $f'(x)$ is bounded, $f(a)=f(b)$, then there exists $c$ ($a<c<b$), which satisfies $f'(c)=0$....
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### $2019f'(x)+2020f(x)\geq2021$

Find all continuous function $f:[0,1]\rightarrow\mathbb{R}$ which is differentiable on $(0,1)$ and $$f(0)=f(1)=\frac{2021}{2020}\textrm{ while }2019f'(x)+2020f(x)\geq2021,\forall x\in(0,1).$$ The ...
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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'(... 2 votes 1 answer 46 views ### How do I determine further solutions of the equation using Rolle's theorem? I gave this equation 2^x=1+x^2 with the 1st zero is x_1=0 and the 2nd zero is x_2=1. (easy reading) Now I want to calculate more zeros using Rolle's theorem, and I've rearranged the function ... 0 votes 1 answer 44 views ### 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 ... 0 votes 0 answers 47 views ### 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 ... 0 votes 1 answer 36 views ### 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 ... 1 vote 0 answers 115 views ### 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 ... 0 votes 0 answers 20 views ### Is it possible to show the following without the assumption of continuity in a closed inteval? Let f(x) be diffrentiable at (a,b) for a,b\in \mathbb{R}. given that f'(x)\neq 0 for any x\in (a,b) proove that f(x)=0 for not more than one point in (a,b) Well, If I was given that f(... 4 votes 1 answer 71 views ### Problem involving Rolle's Theorem Prove that, between any 2 roots of e^x\sin x = 1, there exists a root of e^x\cos x + 1=0. I am able to solve the problem in this way: Let a,b be the roots of f(x)=e^x\sin x - 1. Define g(x)... 4 votes 1 answer 76 views ### Existence of x_0 \in (0,1), s.t. f'(x_0)+1/x_0 f(x_0)=2 f(x) is differentiable on [0,1], \int_{0}^{1}{f(x)}dx=1/2, how to imply exist x_0 \in (0,1), s.t. f'(x_0)+1/x_0 f(x_0)=2 I think it is equal to prove \exists x_0, s.t. (x_0f(x_0))'=2x_0 I ... 0 votes 3 answers 90 views ### Prove there exists a point \xi so that f'(\xi)=0 If we have a function f(x) is differentiable in \mathbb{R} and \lim_{x\rightarrow \pm \infty} f(x) = 17 . How can we show that \exists \xi \in (-\infty, \infty) | f'(\xi) = 0? I'm wondering ... 1 vote 0 answers 38 views ### Mean value theorem to prove existence of zero for second derivative [duplicate] I'm going to try to solve the problem presented below. But as you will realize, I'll run into a problem. I hope you can give me somet tips. Let f(x) be a function which is contiuous on [a, b] and ... 2 votes 1 answer 146 views ### Let f be a continuous function in [0,5] and twice differentiable function in (0,5) such that f(4)=f(5)=0. Prove the following Let f be a continuous function in [0,5] and twice differentiable function in (0,5) such that f(4)=f(5)=0. Prove the following: There exists some a in [0,5] such that nf(a)+af'(a)=0, n\in ... 2 votes 1 answer 77 views ### If P_n(x) has n real roots, then it's derivative has no complex roots Usually, I would use Rolle's Theorem. Every polynomial is continuous and differentiable. Using Rolle's Theorem, derivatives of functions have n-1 real roots: Between every two consecutive roots a ... 2 votes 1 answer 65 views ### Prove that the polynomial has exactly 2 real roots by IVT or Rolle's Theorem I'm trying to prove this by IVT or Rolle's Theorem. Usually if it would say "Prove it has only 1 real root", I would assume it had 2 roots, take the derivative and if it was ≠0, then I ... 2 votes 1 answer 102 views ### Solving the equation of the type g\left( x \right) = \int\limits_0^x {f\left( t \right)dt}  Let f:\left[ {0,1} \right] \to R be a differentiable function. Let g\left( x \right) = \int\limits_0^x {f\left( t \right)dt}  with g\left( 1 \right) = 0 . Which of these equations must have at ... 0 votes 1 answer 33 views ### Interval in which roots of given function lies is: Let f(x)=f\left(x\right)=\frac{x}{\sin x} & x∈\left(0,\frac{\pi}{2}\right), Then prove that the interval in which at least one root of equation h\left(x\right)= \frac{2}{x-f\left(\frac{\pi}{... 1 vote 2 answers 76 views ### If f is twice differentiable such that f\left(\frac{1}{n}\right)=0, then what can we say about f''(0)? The continuity of f, f' and Rolles theorem implies that f(0)=0 and f'(0)=0, but is it true that f''(0). We have a sequence (x_n) such that f''(x_n)=0 and x_n \to 0. But continuity of f''... 0 votes 0 answers 57 views ### f(x) = (x^2 -1)^2(a_0 x^3+a_1 x^2 + a_2 x + a_3 ) f'(x) has exactly 3 distinct real roots and f''(x) has two distinct real roots. Find f(x). Can we get a function in the following form and properties ? f(x) = (x^2 -1)^2(a_0 x^3+a_1 x^2 + a_2 x + a_3 ) f'(x) has exactly 3 distinct real roots and f''(x) has two distinct real roots. ... 0 votes 2 answers 90 views ### Verify Rolle's theorem Verify Rolle's theorem for the function f(x)=x^2+5x-6 in the interval (-6,1) After checking continuous for the function. I saw the function is continuous and differentiable. After checking ... 0 votes 1 answer 74 views ### Rules of checking differentiability for Rolle's theorem I was doing some exercises of Rolle's theorem. But, they didn't check the differentiability the way we checked differentiability normally. I am giving some examples. When I was checking ... 1 vote 1 answer 92 views ### Is Rolle's theorem and Mean values theorem same? You are correct in saying that these theorems are essentially the same. The mean value theorem is a general form of the Roll's theorem where the slope of secant is not necessarily zero. Both theorems ... 0 votes 0 answers 58 views ### What's the meaning of different braces? Since the function f(x) is continuous in the closed interval [a,b] and differentiable in the open interval (a,b), so the function F(x) will be continuous in the interval [a,b] and satisfied ... 1 vote 0 answers 44 views ### Range of values of 'a' so that the function has local maxima/minima at given values of x Find the set of all the possible values of a for which the function:$$f(x)=5+(a-2)x+(a-1)x^2-x^3$$Has a local minimum value at some x<1 and local maximum value at some x>1. I started by ... 1 vote 0 answers 73 views ### Let f(x) be a function such that f(x)=f(4-x) , then find the minimum number of roots of f''(x)=0 in [0,4]. Let f(x) be a non-constant twice differentiable function defined on (-\infty,\infty) such that f(x)=f(4-x) and f(x)=0 has at least two distinct repeated roots in (2,4), then find the minimum ... 1 vote 1 answer 53 views ### If f(a) = f(b) and f'(a) = f'(b), prove that for every real number λ the equation f''(x) − λ(f'(x))^2 = 0 has at least one solution in (a,b) Let f : [a, b] → R be a function, continuous on [a, b] and twice differentiable on (a, b). If f(a) = f(b) and f'(a) = f'(b), prove that for every real number λ the equation f''(x) − λ(f'(... 0 votes 0 answers 28 views ### Upper limit of differential connection to original function I have a homework with a question that goes like this: Let f(x) be a differentiable function at [a,b]. assuming: f(a)=f(b)=0 f(x)>0 for all x in (a,b) there is a M that: |f^′(x)|\le M... 1 vote 1 answer 66 views ### Intuitive approach towards the auxiliary function of Lagrange's mean value theorem proof I have often seen many people using another auxiliary function g(x) such that g(x)=f(x)-\frac{f(b)-f(a)}{b-a}\cdot x where f(x) is our original function continuous in [a,b], differentiable in ... -1 votes 1 answer 61 views ### Change in the number of positive zeros of a continuous function. Let f(x) be any continuous function, then is it true that$$Z^{+}\left(\alpha f(x)+(x+\beta)f'(x)\right)\leq Z^{+}\left(f'(x)\right)+1$$where \alpha>1 is a real number and \beta is any ... 0 votes 2 answers 60 views ### Show that p(D) f(x) has at least one zero on [a, b], where D denotes \frac{d}{dx}. Let p(x) be a real polynomial of degree n with leading coefficient 1 and all roots real. Let R be the reals and f : [a, b] → R be an n times differentiable function with at least n + 1 distinct ... -1 votes 1 answer 68 views ### Is there a conflict between an Euler's ODE and Rolle's theorem here? [closed] Let a polynomial f(x) of degree n>2 or more has n number of distinct non-zero roots then we can prove that the equation$$x^2f''(x)+3xf'(x)+f(x)=0~~~~(1)$$has at least n real root. Let g(... 1 vote 3 answers 131 views ### Comparing e^{4}-2 and 50 without calculator I'm required to compare e^{4}-2 and 50 without using calculator. I thought of the following way: Let a function h(x) be defined as e^{x}-13x. If I can prove that this function at x=4 is ... 0 votes 1 answer 125 views ### Absolute of Second Derivative of f is less than or equal to M Let f:\mathbb{R}\rightarrow\mathbb{R} be a twice differentiable function Suppose |f''(x)| \leq M on [a, b]. Show that for all x \in [a, b]$$|f'(x)| \leq \left|\frac{f(b)-f(a)}{b-a}\right|+(b-...
Assume that $f(x)$ is continuous in $[a,b]$ and differential in $(a,b)$, and $f(a)=f(b)=0$, prove that there exists a $\xi\in(a,b)$, s.t $f'(\xi)=f(\xi)$.