Let f attain its maximum on the set {x R³ : ||x|| = 1} at
the point x' = (x,y,z). Then ||x'|| = 1 that is
Case 1;
.
Then x'= g(x,y) f(x') =
f(g(x,y)) = (f•g)(x,y). Therefore the maximum of f on {x
R³ :
||x|| =1} is same as the maximum of f•g on { x
R² : ||x|≤1}.
Case 2;
.
Then x'= h(x,y) f(x') =
f(h(x,y)) = (f•h)(x,y). Therefore the maximum of f on {x
R³ :
||x|| =1} is same as the maximum of f•h on {x
R² : ||x||≤1}.
Therefore the maximum of f on {x R³ : ||x|| = 1} is
either the maximum of f•g or the maximum of f•h on {x
R² : ||x|| ≤
1}.
2. Define a function g: R3 +R by g(x, y, z) = 2x2 + y2 + x2 + 2xz – 2y – 4. (a) Find all the critical points of g. (b) Compute the Hessian H, of g. (c) Classify the critical points of g. (d) The surface g(x, y, z) = 0 is an ellipsoid . Use the method of Lagrange multipliers to find the maximum value of the function (5 marks) (5 marks) (5 marks) f(x, y, z)...
12. Let g(x), h(y) and p(z) be functions and define f(x, y, z) = g(x)h(y)p(2). Let R= = {(x, y, z) E R3: a < x <b,c sy <d, eszsf} where a, b, c, d, e and f are constants. Prove the following result SS1, 5100,2)AV = L*()dx ["Mwdy ['Plzdz.
Question 3. Consider the function h: R3 → R h(x, y, 2) = (x2 + y2 + 2) +3/(x2 + 2xy + y) (a) What is the maximal domain of h? Describe it in words. (it may help to factor the denominator in the second term) > 0 for any a, (b) It is difficult to immediately find the range of h. Using the fact that a show that h cannot take negative values. Can h be an onto function?...
(a) By the Heine-Borel Theorem, show that R2 is not compact and
the
sphere
S2 ={(x,y,z)∈R3 :x2 +y2 +z2 =1}
is compact in R3.
(b) Show that R2 and S2 is not homeomorphic. (i.e. no continuous
bi-
jective function f between R2 and S2 such that the inverse function
f−1 is continuous).
Question 1. (2 marks) (a) By the Heine-Borel Theorem, show that R2 is not compact and the sphere is compact in R3. (b) Show that R2 and S2...
b) Let a R3 be a vector of length 1. Define H={x E R3 : a·x=0). Here a x denotes the dot product of the vectors a and x. (i) Show that H is a subgroup of R (ii) For λ E R, show that : a·x= is a coset of H in R3. (ii) Is H cyclic? Prove or disprove.
b) Let a R3 be a vector of length 1. Define H={x E R3 : a·x=0). Here a x...
Define T : R3 → R2 by T(x,y,z) = (2x +4y +3z,6x) Show that T is linear.
10. Define φ : R2 → R by φ(x,y-x + y for (x,y) E R2. Show that φ is an onto homomorphism and find the kernel of φ (10 Points)
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2 If(x,y)| 〈 M(x2 + y2)· for all (a·y) E R2 Prove that:
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2...
5. Let F(x, y, z) = (yz, xz, xy) and define 2 Crin = {(x,y,z) : x2 + y2 = r2, 2 = h} Show that for any r > 0 and h ER, le F. dx = 0 Crih
5. (7 points) Let f: R3 → R be the function f(x,y,z) = x2 + y2 +3(2-1)2 Let EC R3 be the closed half-ball E = {(x, y, z) e R$: x² + y2 +< 9 and 2 >0}. Find all the points (x, y, z) at which f attains its global maximum and minimum on E.