Newton's method chooses its own step size by dividing the slope by the curvature:
xn+1=xn−f′(xn)f′′(xn)x_{n+1} = x_n - \frac{f'(x_n)}{f''(x_n)}xn+1=xn−f′′(xn)f′(xn)
Apply exactly one step to
f(x)=x3−6x2+9xf(x) = x^3 - 6x^2 + 9xf(x)=x3−6x2+9x
starting from x0=0x_0 = 0x0=0, then compare f(x1)f(x_1)f(x1) with f(x0)f(x_0)f(x0).
What happens?
Select all that apply.
x1=−0.75x_1 = -0.75x1=−0.75, and fff decreases
x1=0.75x_1 = 0.75x1=0.75, and fff decreases
x1=0.75x_1 = 0.75x1=0.75, and fff increases — the step has moved toward a local maximum
x1=−1.8x_1 = -1.8x1=−1.8, and fff decreases