Question

The root of the equation e power x=4x lies between__

Answer 1

14209

wyieueue

rww

wueoe

w

Answer 2

Given:

$f(x)=e^{x} -4x$

To find: The root of the equation e power x=4x lies between ______.

Solution:

Solve the given expression using Newton-Raphson method.

$\[\begin{array}{l}f\left( x \right) = {e^x} - 4x\\ \Rightarrow f'\left( x \right) = {e^x} - 4\\{x_{n + 1}} = {x_n} - \frac{{f\left( x \right)}}{{f'\left( {{x_n}} \right)}}\end{array}\]$

Use Lambert W function.

$\[\begin{array}{l}{e^x} = 4x\\ \Rightarrow {e^{ - x}} = \frac{1}{{4x}}\\ \Rightarrow - x{e^{ - x}} = - \frac{1}{4}\end{array}\]$

Solve for x.

$\[ \Rightarrow x = - {W_n}\left( { - \frac{1}{4}} \right),\frac{1}{e} < - \frac{1}{4} < 0\]$

Understand that, there are two real solutions. The others are complex-valued.

Therefore, the two real-valued answers are,

$\[x = - {W_0}\left( { - \frac{1}{4}} \right) \approx - 0.357403\]$ and $\[x = - {W_{ - 1}}\left( { - \frac{1}{4}} \right) \approx - 2.15329\]$