Final answer:
The bisection method involves halving the interval between two values to find the root of a function by checking where the function changes sign. After six iterations, the midpoint gives a refined approximation of the root, accurate to five decimal places.
Step-by-step explanation:
The question involves using the bisection method to approximate the solution to the function f(x) = e⁻ᴇ⁴ − ln(x) on the interval [1,2]. To perform the sixth iteration of the bisection method, one would start by checking the sign of f(x) at the endpoints of the interval and then successively halve the interval, choosing the subinterval where the sign of f(x) changes. This process is repeated until the sixth iteration is reached.
By applying the bisection method, we repeatedly narrow down the interval where the root of the function must lie. After each iteration, we check the value of f(x) at the midpoint of the current interval. Since the function changes sign over an interval containing a root, we select the new interval to be the half in which the sign change occurs. The midpoint of the interval used in the sixth iteration, rounded to five decimal places, will give us the required approximation of the root.