$f:[1,3] \rightarrow R$ is a function defined as $f(x)=x^3+a x^2+b x$. If $f(1)-f(3)=0$ and $f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0$, then $a-b$ is equal to

Verify Rolle's theorem for the function $y=x^{2}+2$ on the interval $[-2, 2]$.

The constant $c$ of Lagrange's mean value theorem for $f(x)=\cos x-\sin 2x$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is

Consider the function $f(x) = 8x^2 - 7x + 5$ on the interval $[-6, 6]$. The value of $c$ that satisfies the conclusion of the Mean Value Theorem is:

Let $f(x)$ be a function continuous on $[1, 2]$ and differentiable on $(1, 2)$ satisfying $f(1) = 2, f(2) = 3$ and $f'(x) \geq 1$ for all $x \in (1, 2)$. Define $g(x) = \int_1^x f(t) \, dt$ for all $x \in [1, 2]$. Then the greatest value of $g(x)$ on $[1, 2]$ 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: