348 CHAPTER 18. OPTIMIZATION

12

s2 (vT (H (x+tsv)−H (x))v)

which implies

f (x+sv) = f (x)+12

s2λ

2 |v|2 + 12

s2 (vT (H (x+tsv)−H (x))v)

≥ f (x)+14

s2λ

2 |v|2

whenever s is small enough. Thus in the direction v the function has a local minimum atx. The assertion about the local maximum in some direction follows similarly. ■