Determine whether the following statements are true or false. No justificationis required.(a) Let 2 be a subset of R". If ô 30, then N is open.(b) If f : IR? R is differentiable at (0,0), then f is continuous at (0,0).(c) If f : R? → R², g : R2 → R² are both continuous at (0, 0), then the compositionfog is continuous at (0,0).(d) If two vectors u, v E R° satisfy u v = 0 and u x v = 0, then either u 0 or v = 0.(e) Let f : R → R. If all first order partial derivatives of f at a point a e R' are zero,then any directional derivative of f at a is zero.(f) Let f(x, y, z) be a Coo function on R. If its second order Taylor polynomial at(0,0,0) isP2(x, y, z) = 1+x² + y? + z – xy - yz,%3Dthen f(x, y, z) has a local minimum at (0,0,0).

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Determine whether the following statements are true or false. No justification is required. (a) Let 2 be a subset of R". If N nN=Ø, then 2 is open. (b) If f: R? → R is differentiable at (0, 0), then ƒ is continuous at (0,0). (c) If f : R? → R², g : R² → R² are both continuous at (0,0), then the composition fog is continuous at (0,0). p3 sotisfy u:y O and u x v = 0. then cither u = 0 or v = 0. a) Tf + wo veotore

Determine whether the following statements are true or false. No justification is required. (a) Let 2 be a subset of R". If N nN=Ø, then 2 is open. (b) If f: R? → R is differentiable at (0, 0), then ƒ is continuous at (0,0). (c) If f : R? → R², g : R² → R² are both continuous at (0,0), then the composition fog is continuous at (0,0). p3 sotisfy u:y O and u x v = 0. then cither u = 0 or v = 0. a) Tf + wo veotore

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.2: Divisibility And Greatest Common Divisor

Problem 23E

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Concept explainers

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as:

Question

Determine whether the following statements are true or false. No justification
is required.
(a) Let 2 be a subset of R". If ô 30, then N is open.
(b) If f : IR? R is differentiable at (0,0), then f is continuous at (0,0).
(c) If f : R? → R², g : R2 → R² are both continuous at (0, 0), then the composition
fog is continuous at (0,0).
(d) If two vectors u, v E R° satisfy u v = 0 and u x v = 0, then either u 0 or v = 0.
(e) Let f : R → R. If all first order partial derivatives of f at a point a e R' are zero,
then any directional derivative of f at a is zero.
(f) Let f(x, y, z) be a Coo function on R. If its second order Taylor polynomial at
(0,0,0) is
P2(x, y, z) = 1+x² + y? + z – xy - yz,
%3D
then f(x, y, z) has a local minimum at (0,0,0).

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