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

If the function $f(x) = ax^3 + bx^2 + 11x - 6$ satisfies the conditions of Rolle's theorem for the interval $[1, 3]$ and $f'\left( 2 + \frac{1}{\sqrt{3}} \right) = 0$,then the values of $a$ and $b$ are respectively

$A$ function $f$ is defined by $f(x)=2+(x-1)^{2/3}$ on $[0,2]$. Which of the following statements is incorrect?

If $f(x) = \sin^2 x + x \sin 2x \log x$,then $f(x) = 0$ has

If $f(x) = (x - 1)(x - 2)(x - 3)$ for $x \in [0, 4]$,then the value of $c \in (0, 4)$ satisfying Lagrange's mean value theorem is

Let $f(x)$ be continuous on $[0,4]$, differentiable on $(0,4)$, $f(0)=4$ and $f(4)=-2$. If $g(x)=\frac{f(x)}{x+2}$, then the value of $g^{\prime}(c)$ for some Lagrange's constant $c \in (0,4)$ is