The Newton’s method for finding the maximum of a function involves finding the roots of its derivative. In this case, the derivative of 𝑓(𝑥) is 𝑓′(𝑥) = −9𝑥5 − 8𝑥3 + 12.

Starting with an initial guess of x0 = 2, we can use the Newton’s method formula:

x1 = x0 - 𝑓′(𝑥0)/𝑓′′(𝑥0)

where 𝑓′′(𝑥) is the second derivative of 𝑓(𝑥), which is 𝑓′′(𝑥) = −45𝑥4 − 24𝑥2.

Plugging in the values, we get:

x1 = 2 - (−9(2)5 − 8(2)3 + 12)/(−45(2)4 − 24(2)2) x1 = 2 - (−576)/(−360) x1 = 2 + 1.6 x1 = 3.6

Therefore, the optimum value at the end of the first iteration is x1 = 3.6. None of the given options match this value, so the answer is none of the above.