Verify Rolle's theorem for the function f ( x) = x^ 2 - 5x + 4 on… - Vidyadip

Question

Verify Rolle's theorem for the function $f ( x) = x^{2} - 5x + 4\text{ on} [ 1, 4].$

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Answer

$\text{f (x)} = x^{2} - 5x + 4 \text {( a polynomial function)}$$\text{(i) the function is continuous on [1,4] }$
$\text{(ii) It is differentiable on (1,4)}$
$\text{(iii) f (1)} = \text{f (4)} = 0$
$\therefore$ All the conditions of Rolles' Theoremare satisfied.
$\therefore \text{f' (c)} = 0 \Rightarrow \text{2 c - 5} = 0 \Rightarrow \text{c} = \frac{5}{2}$
$\text{As c} = \frac{5}{2} \in (1,4),$ the Rolle's theorem is verified.

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