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

If the function $f(x) = 2x^2 + 3x + 5$ satisfies the Lagrange's Mean Value Theorem $(LMVT)$ at $x = 3$ on the closed interval $[1, a]$,then the value of $a$ is equal to:

If $f(x) = \cos x$ for $0 \le x \le \frac{\pi}{2}$,then the real number $c$ of the Mean Value Theorem is:

Let $f$ be a function which is differentiable for all real $x$. If $f(2) = -4$ and $f^{\prime}(x) \geq 6$ for all $x \in [2, 4]$, then which of the following is true?

Let $f :[0,1] \rightarrow R$ be a twice differentiable function in $(0,1)$ such that $f(0)=3$ and $f(1)=5$. If the line $y=2x+3$ intersects the graph of $f$ at only two distinct points in $(0,1)$,then the least number of points $x \in(0,1)$,at which $f^{\prime\prime}(x)=0$,is $......$

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