For x³+4x²−10=0 in (1,3/2), consider S1: $x_{k+1}=\sqrt{10/(4+x_k)}$ converges for every initial x₀ in that interval. S2: $x_{k+1}=\frac12\sqrt{10-x_k^3}$ diverges for some such x₀. Which statements hold?
Both iteration maps send [1,1.5] into itself. For g₁, $|g_1^\prime|=\sqrt{10}/[2(4+x)^{3/2}]\le0.142<1$. For g₂, $|g_2^\prime|=3x^2/[4\sqrt{10-x^3}]\le0.656<1$. Both are contractions, so both converge for all supplied starts. S1 is true and the divergence claim S2 false. **A**.
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