Proving inequalities via the mean value theorem

Prove the following inequalities:

Using Lagrange's formula, prove the inequality $n(b-a)a^{n-1} < b^{n}-a^{n} < n(b-a)b^{n-1}$ for $b > a > 0$, $n > 1$.

Using Lagrange's formula, prove the inequality $\dfrac {x}{1+x} < \ln (1+x) < x$ for $x > 0$.

Using Lagrange's formula, prove the inequality $e^{x} > 1+x$.

Using Lagrange's formula, prove the inequality $e^{x} > ex$ for $x > 1$.

Using Lagrange's formula, prove the inequality $x^{\delta }\left|\ln x\right| < \dfrac {1}{\delta e}$ for $0 < x < 1$, $\delta > 0$.

Prove the theorem: if at $x=0$ the functions $\varphi (x), \varphi '(x), \ldots , \varphi^{(n-1)}(x)$ vanish, and the function $\varphi^{(n)}(n)$ is positive for $x > 0$, then $\varphi (x) > 0$ for positive $x$.

Prove the theorem: if

and $\varphi^{(n)}(x) > \psi^{(n)}(x)$ for $x > 0$, then for positive $x$ the inequality $\varphi (x) > \psi (x)$ holds.

Prove the inequality $x-\dfrac {x^{3}}{6} < \sin x < x$, for $x > 0$.

Prove the inequality $x-\dfrac {x^{3}}{3} < \arctg x < x$, for $x > 0$.

Prove the inequality $\tg x > x+\dfrac {x^{3}}{3}$, for $0 < x < \dfrac {\pi }{2}$.

Prove the inequality $x-\dfrac {x^{2}}{2} < \ln (1+x) < x$; $x > 0$.

By direct squaring, prove the inequalities

Prove that for $0 < x < 1$ the quantity $e^{2x}$ is less than $\dfrac {1+x}{1-x}$.

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: 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 Theorem 2.4.20.

Theorem 2.4.20: Let $f$ be a differentiable function on a set $E \subset \mathbb {R}$. Then $f$ is Lipschitz with constant $K = \sup \left\{ \left|f'(x)\right| : x \in E\right\}$ if and only if it has a bounded derivative.