Verify Rolle's theorem for the function $\displaystyle f(x)=10x-x^{2}$ in the interval ;[0,10]

$f\left( x \right) =10x-{ x }^{ 2 }\quad \quad \left[ 0,10 \right]$

$10x-{ x }^{ 2 }$ is continuous in $\left[ 0,10 \right]$ since it is polynomial function.

$f'\left( x \right) =10-2x$ is defined for all values of x in $(0,10)$

$\Rightarrow f\left( x \right)$ is differentiable on $(0,10)$

$f\left( 0 \right) =f\left( 10 \right) =0$

$\therefore$ There exists a $c,a\le c\le b$

$0\le c\le 10$

Such that

$f'\left( c \right) =0$

$10-2x=0$

$x=5$ $5$ lies in $\left[ 0,10 \right]$