Examine the applicability of Mean Value Theorem:
$(i)$ $f(x)=[x]$ for $x \in[5,9]$
$(ii)$ $f(x)=[x]$ for $x \in[-2,2]$
$(iii)$ $f(x)=x^{2}-1$ for $x \in[1,2]$
Verify Mean Value Theorem, if $f(x)=x^{2}-4 x-3$ in the interval $[a, b],$ where $a=1$ and $b=4$
Verify Mean Value Theorem, if $f(x)=x^{3}-5 x^{2}-3 x$ in the interval $[a, b],$ where $a=1$ and $b=3 .$ Find all $c \in(1,3)$ for which $f^{\prime}(c)=0$
If Rolle's theorem holds for the function $f(x) = 2{x^3} + b{x^2} + cx,\,x\, \in \,\left[ { - 1,1} \right]$ at the point $x = \frac{1}{2}$ , then $(2b+c)$ is equal to
Let $y = f (x)$ and $y = g (x)$ be two differentiable function in $[0,2]$ such that $f(0) = 3,$ $f(2) = 5$ , $g (0) = 1$ and $g(2) = 2$. If there exist atlellst one $c \in \left( {0,2} \right)$ such that $f'(c)=kg'(c)$,then $k$ must be
For the function$x + {1 \over x},x \in [1,\,3]$, the value of $ c$ for the mean value theorem is