If the Mean Value Theorem holds for the function $f(x)=(x-1)(x-2)(x-3)$ on the interval $x \in [0, 4]$,then the values of $c$ as per the theorem are:

If $f(x) = x^{\alpha} \log x, x > 0, f(0) = 0$ and $f(x)$ satisfies Rolle's theorem on $[0, 1]$,then what is the value of $\alpha$?

If $2a + 3b + 6c = 0$,then at least one root of the equation $ax^2 + bx + c = 0$ lies in the interval:

Let $f: R \rightarrow R$ be a thrice differentiable function such that $f(0)=0, f(1)=1, f(2)=-1, f(3)=2$ and $f(4)=-2$. Then,the minimum number of zeros of $(3 f^{\prime} f^{\prime \prime} + f f^{\prime \prime \prime})(x)$ is....................

If $2a + 3b + 6c = 0$,then at least one root of the equation $ax^2 + bx + c = 0$ lies in which interval?

Let $f(x)$ be a differentiable function in $[2,7]$. If $f(2)=3$ and $f^{\prime}(x) \leq 5$ for all $x$ in $(2,7)$, then the maximum possible value of $f(x)$ at $x=7$ is