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

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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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 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).
Transcribed Image Text: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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