First slide

Mean value theorems

Question

$The equation e^{x - 1} + x - 2 = 0 as$

Moderate

A

) one real root (2)
(3) (4)
B

two real roots
C

three real roots
D

four real roots

Solution

Clearly, x = 1 satisfies the given equation. Assume that $f (x) = e^{x - 1} + x - 2 = 0$
has a real root $α$ other than x = 1.
$We may suppose that α > 1 (the case α < 1 is exactly similar).$
$Applying Rolle's theorem on [1, α] (if α < 1 apply the$
$theorem on [α, 1], we get β$
$\begin{array}{l} \in (1, α) such that f^{'} (β) = 0 . But f^{'} (β) = e^{β - 1} + 1, \\ so that e^{β - 1} = - 1, which is not possible. Hence there is noreal root other than 1. \end{array}$

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