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Question

Verify the Rolle's theorem for the function $$\displaystyle f(x)=x^{2}-3x+2$$ on the interval[1,2]


A
No Rolle's theorem is not applicable in the given interval
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B
Yes Rolle's theorem is applicable in the given interval and the stationary point x=54
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C
Yes Rolle's theorem is applicable in the given interval and the stationary point x=32
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D
nnone of these
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Solution

The correct option is B Yes Rolle's theorem is applicable in the given interval and the stationary point $$x=\frac { 3 }{ 2 } $$
It can be easily seen that $$f(x)=x^{2}-3x+2$$ is continuous as differentiable on R (being a polynomial) $$\Rightarrow f(x)$$ is continous in (1,2) and differentiable in [1,2]. Also, we have
$$f(1)=f(2)=0$$.
 Thus, $$f(x)$$ satisfies all the conditions of Rolle's theorem in $$[1,2]$$ $$\Rightarrow \displaystyle \exists$$ at least one number, say $$x$$ in $$[1,2]$$ such that $$\displaystyle
f^{'}(c)=0.$$ Now, $$\displaystyle f^{'}(x)=2x-3=0\Rightarrow x=\frac{3}{2}$$ Since, the root (stationary point)  $$\displaystyle x=\frac{3}{2}$$ lies in the interval(1,2). 
Hence Rolle's theorem is verified.

Mathematics

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