Root of the function f(x)=xe^x−R

Roderick Bradley

Roderick Bradley

Open question

2022-08-16

Root of the function f ( x ) = x e x R

Answer & Explanation

Gauge Howard

Gauge Howard

Beginner2022-08-17Added 19 answers

Newton's method will work fairly well here. In particular, letting f ( x ) = x e x R, then f ( x ) = x e x + e x , so our iteration step will be
x n + 1 = x n x n e x n R ( x n + 1 ) e x n
So if we choose, for example, R = 1 and x 0 = 0, we find that
x 0 = 0 x 1 = 1 x 2 0.6839 x 3 0.5775 x 4 0.5672 x 5 0.5671433 x 6 0.5671433
So it converges pretty quickly. Checking this in our original function, we find that
f ( 0.5671433 ) 2.22 × 10 16

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