2. Let f : R" → R be a continuously differentiable function, while a E R" is its non-stationary point (i.e., Vf(x) # 0). Moreover, let d* be the vector defined by Vf(x) ||Vf(x)||' d* = - while d is...

0. (ii) For t z 0, approximate f(x+td*) and f(x+td) using the first-order Taylor polynomial of f(x) at x. "/>
Extracted text: 2. Let f : R" → R be a continuously differentiable function, while a E R" is its non-stationary point (i.e., Vf(x) # 0). Moreover, let d* be the vector defined by Vf(x) ||Vf(x)||' d* = - while d is any unit vector in R" different from d* (i.e., d e R", ||d|| = 1, d + d*). Show that there exist real numbers d E (0, 00), ɛ E (0, 0) such that %3D f(x + td*) – f(x + td) < -ôt="" (1)="" for="" all="" t="" e="" [0,="" ɛ).="" remark:="" this="" question="" is="" a="" slightly="" extended="" version="" of="" lemma="" 6.1,="" chapter="" i.2.="" hints:="" (i)="" select="" &="" as="" 8="-;" (vf(x))"="" (d*="" –="" d)="" and="" use="" the="" cauchy-schwartz="" inequality="" to="" show="" 8=""> 0. (ii) For t z 0, approximate f(x+td*) and f(x+td) using the first-order Taylor polynomial of f(x) at x.