It is given that
f(x)=ex+2+x−10f(x) = e^{x+2} + x - 10f(x)=ex+2+x−10
Show that the equation f(x)=0f(x) = 0f(x)=0 can be written in the form x=ln(10−x)−2x = \ln(10 - x) - 2x=ln(10−x)−2.
Use the iteration formula
xn+1=ln(10−xn)−2x_{n+1} = \ln(10 - x_n) - 2xn+1=ln(10−xn)−2
with x0=0.5x_0 = 0.5x0=0.5 to find, to three decimal places, the values of x1x_1x1, x2 x_2\,x2 and x3x_3x3.
The iteration function is g(x)=ln(10−x)−2g(x)=\ln(10-x)-2g(x)=ln(10−x)−2.
Show that this iteration converges to the root for values of x x\,x close to 0.275.