Consider the function 𝑓(𝑥) = 𝑥! sin 𝑥 − 𝑥𝑒-x where 0 ≤ 𝑥 ≤ 1. (a

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Consider the function 𝑓(𝑥) = 𝑥! sin 𝑥 −
𝑥𝑒-x where 0 ≤ 𝑥 ≤ 1.
(a) Suppose the golden section method is used to estimate the value
of 𝑥 in the interval [0, 1]
where the function 𝑓(𝑥) is minimized with an absolute error less
than 0.1. How many
iterations are needed? Estimate the value of 𝑥 at the minimum in
the interval [0, 1] using the
golden section method up to the required accuracy. Write your
estimate correct to 3 decimal
places.
(b) Approximate the value of 𝑥 at the minimum using the secant
method. Use 𝑥0 = 0, 𝑥1 = 1.
What are the estimated values of 𝑥 in the first three iterations
(𝑥2, 𝑥3 and 𝑥4)? Round your
answers to 3 decimal places.
(c) Approximate the value of 𝑥 at the minimum using the Newton’s
method. Use 𝑥0 = 1. What
are the estimated values of 𝑥 in the first three iterations
(𝑥1, 𝑥2 and 𝑥3)? Round your
answers
to 3 decimal places.