Question

use the intermediate value theorem to show that there is a root of the given equation in the specified interval.
( e ^ { x } = 5 - 4 x , quad ( 0,1 ) )
the equation ( e ^ { x } = 5 - 4 x ) is equivalent to the equation ( f ( x ) = e ^ { x } - 5 + 4 x = 0 ). ( f ( x ) ) is continuous on the interval ( 0,1 ). ( f ( 0 ) = ), and ( f ( 1 ) = ). since
, there is a number ( c ) in ( ( 0,1 ) ) such that ( f ( c ) = 0 ) by the intermediate value theorem. thus, there is a root of the equation ( e ^ { x } = 5 - 4 x ), in the interval

Explanation:

Step1: Calculate \(f(1)\)

Given \(f(x)=e^{x}-5 + 4x\), substitute \(x = 1\) into \(f(x)\).

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Since \(e\approx2.718\), then \(f(1)=e - 1\approx2.718-1 = 1.718\)

Answer:

\(f(1)\approx1.718\)