Newton's method for optimisation updates a parameter with
wt+1=wt−f′(wt)f′′(wt)w_{t+1} = w_t - \frac{f'(w_t)}{f''(w_t)}wt+1=wt−f′′(wt)f′(wt)
Apply it to
f(w)=w3−6w2+9w+2f(w) = w^3 - 6w^2 + 9w + 2f(w)=w3−6w2+9w+2
starting from w0=4w_0 = 4w0=4.
Compute w2w_2w2, then identify and classify the critical point this iteration is converging to.
Select all that apply.
w2=3.25w_2 = 3.25w2=3.25; it is converging to w=3w = 3w=3, a local minimum.
w2=3.025w_2 = 3.025w2=3.025; it is converging to w=3w = 3w=3, a local maximum, since f′′(3)=6>0f''(3) = 6 > 0f′′(3)=6>0.
w2=3.025w_2 = 3.025w2=3.025; it is converging to w=3w = 3w=3, a local minimum, since f′′(3)=6>0f''(3) = 6 > 0f′′(3)=6>0.
w2≈2.32w_2 \approx 2.32w2≈2.32; it is converging to a point where fff itself is zero, not to a critical point.