Solve example on various calculators

[carey]

(12-10-2023 04:32 AM)lrdheat Wrote:  The CASIO 9750giii has the same problem in the equation app, but finds the -1 and ~-.41 roots in calculate mode using SolveN and the Solve for a range of 2 to 3 finds the ~2.41 root ( a range encompassing the full range of roots only finds the ~2.41 root).

I am wondering why the SolveN and Solve on the non CAS CASIO 9750giii manage to produce all 3 results, SolveN the -1 and -.41… roots, Solve the 2.41… roots.

Just speculating, but two possibilities come to mind:
1) The range cited (2 to 3) does not encompass the full range of roots -- only the 2.41 root. The range -2 to 3 would encompass the full range of roots.
2) Since Newton's Method can fail (either slow convergence or not at all) if the initial guess or an iterate is near f'(x) = 0 (because the line never intersects the x-axis), the 2.41 root is found using Solve as it is well away from f'(x) = 0, but the other two roots are near the local maximum where f'(x) = 0, so it becomes possible that an iterate occurs near f'(x)=0.

Thank you for this interesting thread!