The value of $c$ for which the Lagrange's Mean Value Theorem $(LMVT)$ is applicable for the function $f(x) = x(x+3)(x-2)$ in the interval $[-1, 4]$ is:

$f:[1,3] \rightarrow R$ 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

If the function $f(x) = ax^2 + bx + \sin x$ satisfies all the conditions of Rolle's theorem on $[0, \pi]$ and the slope of the tangent to the curve $y = f(x)$ at $x = \frac{\pi}{4}$ is zero, then $a - b = \dots$

Suppose $f(x)$ is twice differentiable in the interval $[1, 3]$ and $f(1)=f(3)$. If $|f^{\prime \prime}(x)| \leq 2$,then for all $x$ in $[1, 3]$,which one of the following is true?

For all real values of $a_{0}, a_{1}, a_{2}, a_{3}$ satisfying $a_{0}+\frac{a_{1}}{2}+\frac{a_{2}}{3}+\frac{a_{3}}{4}=0$, the equation $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}=0$ has a real root in the interval

Let $f(1) = -2$ and $f'(x) \ge 4.2$ for $1 \le x \le 6$. The smallest possible value of $f(6)$ is: