Question

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

• A. Yes Rolle's theorem is applicable and the stationary point is $x=5$
• B. Yes Rolle's theorem is applicable and the stationary point is $x=4$
• C. No Rolle's theorem is not applicable in the given interval
• D. none of these

Answer 1

The correct option is A Yes Rolle's theorem is applicable and the stationary point is $x=5$

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

$10x-x^2$ is continuous in $[0,10]$ since it is polynomial function.

$f^{\prime }\left(x\right)=10-2x$ 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

$f^{\prime }\left(c\right)=0$

$10-2x=0$

$x=5$ $5$ lies in $[0,10]$