Mean Value Theorems for Differentiable Functions

Verify the validity of Rolle's theorem for the function $x(x^{2}-1)$ on the intervals $[-1 ; 1]$ and $[0 ; 1]$.

On the intervals $(-1 ; 1)$ and $(1 ; 2)$ find the points at which the tangent to the graph of the function $f(x) = \left(x^{2}-1\right)(x-2)$ is horizontal.

On the interval $(0 ; 1)$ find a point $\xi$ such that the tangent to the graph of the function $y = x^{3}$ at the point $(\xi ; \xi^{3})$ is parallel to the chord of the graph joining the points $(0 ; 0)$ and $(1 ; 1)$.

Prove that between two real roots of a polynomial with real coefficients there is a root of its derivative.

Prove that if the function $f$ is differentiable $n$ times on the interval $[a ; b]$ and vanishes at $n+1$ points on it, then there exists $\xi \in (a ; b)$ such that $f^{(n)}(\xi ) = 0$.

Prove that if the function $f$ is differentiable $n$ times on the interval $[a ; b]$, $a = x_{1} < x_{2} < \ldots < x_{k} = b$, $x_{i} \in [a ; b]$, $n_{i}$ are natural numbers, $n_{1}+\ldots +n_{k} = n-1$, and $f^{(j_{i})}(x_{i}) = 0, j_{i} = 0,1, \ldots , n_{i}-1$, $i = 1,2, \ldots , k$, then there exists a point $\xi$ on the interval $[a ; b]$ such that $f^{(n)}(\xi ) = 0$.

Prove that the roots of the derivative of the polynomial

are real, simple, and lie in the intervals $(0 ; 1),(1 ; 2),(2 ; 3)$, $(3 ; 4)$.

Prove that all the roots of the equation $\left(1+x^{2}\right)^{n} \frac{d^{n}\left(1+x^{2}\right)^{-1}}{d x^{n}} = 0$ are real.

Prove that all the roots of the equation $x^{2 n} e^{-1 / x} \frac{d^{n} e^{-1 / x}}{d x^{n}} = 0$ are real.

Prove that all derivatives of a polynomial with real coefficients having only real roots can also have only real roots.

Prove that the roots of all derivatives (of order $m < 2 n$) of the Legendre polynomials

are real, simple, and lie in the interval $(-1 ; 1)$.

Prove that for the Laguerre polynomials

all the roots are positive.

Prove that for the Chebyshev-Hermite polynomials

all the roots are real and lie in the interval $(-\sqrt{2 n+1} ; \sqrt{2 n+1})$.

Applying Lagrange's theorem to the function $1 / x^{\alpha }$, prove that for any $n \in \mathbb {N}$ and any $\alpha > 0$ the inequality holds

Using Lagrange's theorem, prove the inequalities:

Prove that for $x \geqslant 0$ the equality holds

where $1 / 4 \leqslant \theta (x) < 1 / 2$, with $\lim_{x \rightarrow +0} \theta (x) = 1 / 4, \lim_{x \rightarrow +\infty } \theta (x) = 1 / 2$.

Prove: if the function $f$ is differentiable $n$ times for $x \geqslant 0$, $f(0) = f^{\prime }(0) = \ldots = f^{(n-1)}(0) = 0$, and $f^{(n)}(x) > 0$ for $x > 0$, then also $f(x) > 0$ for $x > 0$.

Prove: if the functions $f$ and $g$ are differentiable $n$ times for $x \geqslant 0$ and $f(0) = g(0), f^{\prime }(0) = g^{\prime }(0), \ldots , f^{(n-1)}(0) = g^{(n-1)}(0)$, $f^{(n)}(x) > g^{(n)}(x)$ for $x > 0$, then also $f(x) > -g(x)$ for $x > 0$.

Prove that if the function $f$ is differentiable and unbounded on a finite interval $(a ; b)$, then its derivative is also unbounded on this interval.

Prove that if the function $f$ is differentiable on a finite or infinite interval $(a ; b)$ and there exist equal finite limits, or infinite limits of the same sign, $\lim_{x \rightarrow a+0} f(x)$ and $\lim_{x \rightarrow b-0} f(x)$, then there exists a point $\xi \in (a ; b)$ such that $f^{\prime }(\xi ) = 0$.

Let $f(x)-f(0) = x f^{\prime }(\xi (x)), 0 < \xi (x) < x$. Prove that if $f(x) = x \sin (\ln x)$ for $x > 0$ and $f(0) = 0$, then the function $\xi (x)$ is discontinuous on an arbitrarily small interval $(0 ; \varepsilon ), \varepsilon > 0$.

Prove that if the function $f$ is differentiable for $x > a$ and $\lim_{x \rightarrow +\infty } f^{\prime }(x) = 0$, then $\lim_{x \rightarrow +\infty } \frac{f(x)}{x} = 0$.

Prove that if the function $f$ is differentiable for $x > a$ and $\lim_{x \rightarrow +\infty } \frac{f(x)}{x} = 0$, then $\varliminf_{n \rightarrow +\infty }\left|f^{\prime }(x)\right| = 0$.

Prove that if the function $f$ is differentiable on the interval $[a ; b], a b > 0$, then

where $a < \xi < b$.

Prove that if the function $f$ satisfies the conditions of Rolle's theorem on the interval $[a ; b]$ and is not constant, then there exist points $\xi_{1}$ and $\xi_{2}$ on this interval such that

Prove that if the function $f$ is continuous on the interval $[a ; b]$, differentiable on the interval $(a ; b)$, and is not linear, then there exists a point $\xi \in (a ; b)$ such that

Prove that if the function $f$ is twice differentiable on the interval $[a ; b]$ and $f^{\prime }(a) = f^{\prime }(b) = 0$, then there exists a point $\xi \in (a ; b)$ such that

Let the function $f$ be continuous in some neighborhood of the point $x_{0}$ and differentiable in a punctured neighborhood of this point. Prove that if the limits $\lim_{x \rightarrow x_{0}-0} f^{\prime }(x)$ and $\lim_{x \rightarrow x_{0}+0} f^{\prime }(x)$ exist and are not equal to each other, then the function $f$ is not differentiable at the point $x_{0}$.

Prove that if the function $f$ is differentiable in some punctured neighborhood of the point $x_{0}$, continuous at this point itself, and the finite limit $\lim_{x \rightarrow x_{0}} f^{\prime }(x)$ exists, then the derivative $f^{\prime }\left(x_{0}\right)$ also exists, and $f^{\prime }\left(x_{0}\right) = \lim_{x \rightarrow x_{0}} f^{\prime }(x)$ (in other words, prove Corollary 3 of Lagrange's theorem).

Prove that if the function $f$ is differentiable on the interval $[1 ; 2]$, then there exists a point $\xi \in (1 ; 2)$ such that $f(2)-f(1) = \frac{\xi^{2}}{2} f^{\prime }(\xi )$.

Prove that if the function $f$ is differentiable on the interval $[a ; b]$, with $f(a) = f(b)$, then there exists a point $\xi \in (a ; b)$ such that $f(a)-f(\xi ) = \left(\xi f^{\prime }(\xi )\right) / 2$.

Let the function $f$ be continuously differentiable on the interval $(a ; b)$, and let its derivative be monotonic on this interval. Prove that if $f\left(x_{0}\right) = 0, x_{0} \in (a ; b)$, then $\lim_{x \rightarrow x_{0}} \frac{f(x)}{f^{\prime }(x)} = 0$.

Determine whether there always exists a point $\xi \in (a ; b)$ such that $f^{\prime }(\xi ) = 0$, if the function $f$ satisfies all the conditions of Rolle's theorem except one of the following:

1. the function $f$ is continuous on the interval $[a,b]$;

2. the function $f$ has, at every point of the interval $(a,b)$, a finite derivative or an infinite derivative of definite sign;

3. $f(a) = f(b)$.

Are the conditions of Rolle's theorem necessary and sufficient for there to exist a point $\xi \in (a ; b)$ such that $f^{\prime }(\xi ) = 0$?

Determine whether the Lagrange finite-increment formula holds on the interval $[-1 ; 1]$ for the function

Let the function $f$ be continuously differentiable on the interval $(a ; b)$. Is it true that for any $\xi \in (a ; b)$ there exist points $x_{1} \in (a ; b)$ and $x_{2} \in (a ; b)$ such that

Let the function $f$ satisfy the condition $\left|f(x+h)-f(x-h)\right| < h^{2}$ for any $x \in \mathbb {R}$ and any $h > 0$. Prove that $f(x) =$ const.

Let the function $f$ be differentiable for all $x \in \mathbb {R}$ and let $\lim_{x \rightarrow \infty } f(x) = 0$. Prove that there exists a point $\xi$ such that $f^{\prime }(\xi ) = 0$.

Let the function $f$ be differentiable on the interval $[a ; b]$ and $f^{\prime }(a) f^{\prime }(b) < 0$. Prove that there exists a point $x_{0} \in (a ; b)$ such that $f^{\prime }\left(x_{0}\right) = 0$.

Let the function $f(x)$ have a continuous second-order derivative on $\mathbb {R}$, and suppose finite limits of this function exist as $x \rightarrow -\infty$ and $x \rightarrow +\infty$. Prove that there is a point $x_{0}$ such that $f^{\prime }\left(x_{0}\right) = 0$.

Let the function $f(x)$ be continuous on the interval $[a ; b]$ and differentiable on the interval $(a ; b)$, where $a > 0$. Prove that there exists a point $\xi \in (a ; b)$ such that

Let the function $f(x)$ be continuous on the interval $[0 ; 1]$, differentiable on the interval $(0 ; 1)$, and satisfy the conditions: $f(0) = 5$, $f(1) = 3, f^{\prime }(x) \geqslant -2$ for all $x \in (0 ; 1)$. Prove that $f(x)$ is a linear function on the interval $[0 ; 1]$.

Let the function $f(x)$ be continuous on the interval $[0 ; 2]$, differentiable on the interval $(0 ; 2)$, and satisfy the conditions: $f(0) = -5$, $f(2) = -1 ; f(x) \leqslant -2$ for all $x \in (0 ; 2)$. Prove that $f(x)$ is a linear function on the interval $[0 ; 2]$.

Let the function $f(x)$ be differentiable for $x > 1$ and let a finite $\lim_{x \rightarrow +\infty } f(x)$ exist. Prove that $f^{\prime }(x)$ cannot have a finite nonzero limit as $x \rightarrow +\infty$. Can $f^{\prime }(x)$ fail to have a limit as $x \rightarrow +\infty$?

Let the function $f(x)$ be defined on the interval $(a ; b)$, and suppose that for any points $x_{1}$ and $x_{2}$ from this interval the inequality $\left|f\left(x_{2}\right)-f\left(x_{1}\right)\right| \leqslant \left|x_{2}-x_{1}\right|^{\alpha }$ holds, where $\alpha > 1$. Prove that the function $f(x)$ is constant on the interval $(a ; b)$.