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

Rolle's theorem is true for the function $f(x) = {x^2} - 4 $ in the interval

Let $f(x) = \left\{ {\begin{array}{*{20}{c}}
  {{x^2}\ln x,\,x > 0} \\ 
  {0,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0} 
\end{array}} \right\}$, Rolle’s theorem is applicable to $ f $ for $x \in [0,1]$, if $\alpha = $

If $f$ and $g$ are differentiable functions in $[0, 1]$ satisfying $f\left( 0 \right) = 2 = g\left( 1 \right)\;,\;\;g\left( 0 \right) = 0,$ and $f\left( 1 \right) = 6,$ then for some $c \in \left] {0,1} \right[$  . .

In  $[0, 1]$ Lagrange's mean value theorem is $ NOT$  applicable to

Suppose that $f (0) = - 3$ and $f ' (x) \le 5$ for all values of $x$. Then the largest value which $f (2)$ can attain is