  Question

Verify Rolle's theorem for the function $$\displaystyle f(x)=10x-x^{2}$$ 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  Solution

The correct option is A Yes Rolle's theorem is applicable and the stationary point is $$x=5$$$$f\left( x \right) =10x-{ x }^{ 2 }\quad \quad \left[ 0,10 \right]$$$$10x-{ x }^{ 2 }$$ 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 $$\left[ 0,10 \right]$$Mathematics

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