2.Let f : R" → R be a continuously differentiable function, while a E R" is itsnon-stationary point (i.e., Vf(x) # 0). Moreover, let d* be the vector defined byVf(x)||Vf(x)||'d* = -while d is any unit vector in R" different from d* (i.e., d e R", ||d|| = 1, d + d*). Showthat there exist real numbers d E (0, 00), ɛ E (0, 0) such that%3Df(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 & as8 = -; (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 polynomialof f(x) at x.

Answered: Image /qna-images/answer/ba51000b-c600-4003-86d4-33b854f86c96.jpg

Let f : R" → R be a continuously differentiable function, while æ € R" is its non-stationary point (i.e., Vƒ(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, 0), ɛ e (0, ∞) such that 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 = -(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 polymomial of f(x) at x.

Let f : R" → R be a continuously differentiable function, while æ € R" is its non-stationary point (i.e., Vƒ(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, 0), ɛ e (0, ∞) such that 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 = -(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 polymomial of f(x) at x.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.

Question

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.

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