Let $f(x) = \log(1 + x^2)$ and $A$ be a constant such that $\frac{|f(x) - f(y)|}{|x - y|} \leq A$ for all real $x, y$ where $x \neq y$. Then,the least possible value of $A$ is

Let $f(x)=(x-1)(x-2)(x-3)$,where $x \in [0,4]$. Find the values of $c$ if Lagrange's Mean Value Theorem $(LMVT)$ can be applied.

Let $f(x)$ be continuous on $[0,4]$,differentiable on $(0,4)$,$f(0)=4$ and $f(4)=-2$. If $g(x)=\frac{f(x)}{x+2}$,then the value of $g^{\prime}(c)$ for some Lagrange's constant $c \in (0,4)$ is

For all $x \in [0, 2024]$,assume that $f(x)$ is differentiable,$f(0) = -2$,and $f^{\prime}(x) \geq 5$. Then the least possible value of $f(2024)$ is:

If a polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0 = 0$,where $n$ is a positive integer,has two distinct roots $\alpha$ and $\beta$,then how many roots does the equation $na_nx^{n-1} + (n - 1)a_{n-1}x^{n-2} + \dots + a_1 = 0$ have in the interval $(\alpha, \beta)$?

Let $f:[a, b] \rightarrow R$ be differentiable on $[a, b]$ and $k \in R$. Let $f(a)=0=f(b)$. Also let $J(x)=f'(x)+k f(x)$. Then