Question

Verify Rolle's theorem for the following function on the indicated interval f(x) = x² − 4x + 3 on [1, 3] ?

Answer 1

$$f\left(x\right)=x^2-4x+3$$

We know that a polynomial function is everywhere derivable and hence continuous.
So, being a polynomial function, 

$$f\left(x\right)$$is continuous and derivable on $$[1,3]$$ .

Also,

$$f\left(1\right)={\left(1\right)}^2-4\left(1\right)+3=1-4+3=0$$

$$f\left(3\right)={\left(3\right)}^2-4\left(3\right)+3=9-12+3=0$$

$$\therefore f\left(1\right)=f\left(3\right)=0$$

Thus, all the conditions of Rolle's theorem are satisfied.
Now, we have to show that there exists $$c\in (1,3)$$ such that $$f^{\prime }\left(c\right)=0$$ .

We have

$$f\left(x\right)=x^2-4x+3$$

$$\Rightarrow f^{\prime }\left(x\right)=2x-4$$

$$\therefore f^{\prime }\left(x\right)=0\Rightarrow 2x-4=0\Rightarrow x=2$$

Thus, 

$$c=2\in (1,3)\text{ such that }{f}^{\prime }\left(c\right)=0$$

Hence, Rolle's theorem is verified.