The function $f(x) = {x^3} - 6{x^2} + ax + b$ satisfy the conditions of Rolle's theorem in $[1, 3]. $ The values of  $a $ and $ b $ are

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 $x \in[1,2]$

Examine if Rolle's Theorem is applicable to any of the following functions. Can you say some thing about the converse of Roller's Theorem from these examples?

$f(x)=x^{2}-1$ for $x \in[1,2]$

Examine if Rolle's Theorem is applicable to any of the following functions. Can you say some thing about the converse of Roller's Theorem from these examples?

$f(x)=[x]$ for $x \in[-2,2]$

Examine if Rolle's Theorem is applicable to any of the following functions. Can you say some thing about the converse of Roller's Theorem from these examples?

$f(x)=[x]$ for $x \in[5,9]$

If $f:[-5,5] \rightarrow \mathrm{R}$ is a differentiable function and if $f^{\prime}(x)$ does not vanish anywhere, then prove that $f(-5) \neq f(5).$