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
B
Yes Rolle's theorem is applicable in the given interval and the stationary point x=54
C
Yes Rolle's theorem is applicable in the given interval and the stationary point x=32
D
nnone of these

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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