a Show that the quadratic polynomial P() g1 h1 h3 ( 2) interpolates the function h defined in (10 18) h() g(x(0) g x(0)) at 0, 2, and 3 b Show that a critical point of P occurs at 0 1 2 (2 h1 h3) ...

a We have 1 0 g 1 gx 1 x n gx 0 h a 1 g 3 gx 0 a 3 gx 0 h a 3 g 2 gx 0 a 2 gx 0 h a 2 ...

a. Show that the quadratic polynomial P() = g1 + h1 + h3 ( 2) interpolates

Question:

a. Show that the quadratic polynomial
P(α) = g1 + h1α + h3α (α − α2)
interpolates the function h defined in (10.18):
h(α) = g(x(0) − α∇g x(0))
at α = 0, α2, and α3.
b. Show that a critical point of P occurs at
α0 = 1/2 (α2 − h1 / h3).