If Rolle's theorem holds for the function $f(x)=x^{3}-ax^{2}+bx-4$ on the interval $x \in [1, 2]$ with $f^{\prime}\left(\frac{4}{3}\right)=0$,then the ordered pair $(a, b)$ is equal to

If Rolle's theorem holds for the function $f(x) = 2x^3 + ax^2 + bx$ in the interval $[-1, 1]$ for the point $c = \frac{1}{2}$,then the value of $2a + b$ is

Let $f(x)$ be continuous on $[0, 5]$ and differentiable in $(0, 5)$. If $f(0) = 0$ and $|f^{\prime}(x)| \leq \frac{1}{5}$ for all $x$ in $(0, 5)$, then which of the following is true for all $x$ in $[0, 5]$?

The value of $c$ satisfying the conditions and conclusions of Rolle's theorem for the function $f(x) = x \sqrt{x+6}$ on the interval $x \in [-6, 0]$ is:

Consider $f(x) = |1 - x|$ for $1 \le x \le 2$ and $g(x) = f(x) + b \sin(\frac{\pi}{2}x)$ for $1 \le x \le 2$. Which of the following is correct?

If the equation $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x = 0$,where $a_1 \neq 0$ and $n \ge 2$,has a positive root $x = \alpha$,then the equation $n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \dots + a_1 = 0$ has a positive root,which is: