Question B.3 Consider the minimization problem M(p, y) = min x U(x) s.t. p1 · x1 + ... + pn · xn ≤ y where U : Rn → R is continuous. Prove that the function M(p, y) : Rn + × R+ → R is quasi-concave. [Hint: the subscript + means that all elements of a vector are non negative and at least one is strictly larger than zero.]

Q: The linearization at a = 0 to v8 + 5x is A+ Bx. Compute A and B. A = B = ||

A:

Q: R SECTION 2. 1 1. Compute the gradients of the following functions at the given points. (a) f(x,y)=…

A: Let ft be a C1 function such that f't≠0 Given Fx,y=fx2+y2                      (i) It is known that…

Q: Find a norm of the function f(t) = t² in space C" 0, 1, where norm is given by formula max|f(t)|+…

A: Solution

Q: Define: T:P, → R by T(P) = [ p(x)dx. What is the kernel of T ?

A: Given that: T:P2→RTp=∫01px dx

Q: 6. Suppose that the coefficient functions p(t) and q(t) are continuous in the interval (0, x), and…

A: Use the formula of Wronskian to determine its values. The functions are already given. Determine…

Q: Fit a quadratic function of the form f(t) = co + ct + czť² to the data points (0, 0). (1, 12), (2,…

A:

Q: Give V = C([0, 1]) the inner product defined as follows: (f, g) :| f(t)g(t) dt for all f, g e V.…

A:

Q: 2) Find the homogeneous linear ODE for which the given functions are Basis. -2x -2x , N, = Xe ", y3…

A:

Q: Find all local extrema of the Dirichlet function, defined on (0, 1) by 0, if x is irrational if r =…

A: A function can infinite number of local maxima's and local minima's but there would always be one…

Q: 1. Let f : R° → R³ be f(x, y, z) = (x² + yz, y² + xz, z² + xy). (a) Compute Df(x, y, z). Is f…

A:

Q: The linearization at a = 0 to V6+ 5x is A+ Bx. Compute A and B %3D A = %3D B = %3D

A: Consider the given: Let fx=6+5x    at x=af0=6+50f0=6+0f0=6

Q: Find a norm of the operator A(x) = o (s4 + s? )x(t)dt in space C[0, 1]. a. 4 Ob. O c. 2 4 O d. 15 1…

A:

Q: Consider the linear system x'(t) = 1 x(t) defined for t > 0. Verify that the following (G +) 1+]…

A:

Q: Workout Problem j Part a: ( 1. Find the domain of the vector function r(t) =

A:

Q: Consider the vector-valued function (t - 2)² In(t – 2), sin(t – 2) 12 - 4 12-3t+2 12 +t-6 Find dom R…

A: Here, the domain of R(t) is nothing but an intersection of the domains of each term. Domain of…

Q: Given that vec F =x^ 2 y hat i -(z^ 3 -3x) hat j +4y^ 2 hat k find the k- component of ( nabla* vec…

A:

Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = x2 + y2 + z2…

A: given that a function f(x,y,z)=x2+y2+z2 we need to find the linearizations of f(x,y,z) at (1,1,1).…

Q: Consider the following semilinear first-order PDE ди ди +y ду = xe (i) Find the general solution to…

A:

Q: The linearization at a = 0 to v4 + 4x is A + Bx. Compute A and B. A = B = %3D II

A:

Q: 5) Determine the entire function f(2) = u+ jv satisfying Re(f(2)) = u(x, y) = ae" cos(y) + r? – y²,…

A: That's easy. Thumb up. Have a good day!!! We use Milne Thomsan's theorem

Q: Problem 5.2. Assume that u(0) = u(1) = 0, and that u satisfies 1 [u'v u'v' dx = f fv dx, 0 0 for all…

A: As per the question we are given a functional space V defined as : v ∈ V = { v : ||v|| + ||v'|| <…

Q: Question 9 Show that (a) f (x) = ,x E (0, ∞) and (b) h (r) = e" ,x € R, uniformly continuous on R.…

A:

Q: Question Four (a) Let f(z, y) = r" – 3ry + 18y, where 3r*y-y-6r = 0. Prove that f(r, y) has extrema…

A:

Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A!, 2.11. Ax(1) = x(1)…

A: We have given operator A such that A: C[0,1]-C[0,1].Also we must know that the inverse operator…

Q: (1) Vf (f is a scalar function) 2) ř(t) and r(s), t is time and s is arch length.

A: This is a problem of vector calculus.

Q: 2 Compute L{S, e". cos(t – u)du} a)F(s) %3D (s11)( 1) b)F(s) = (8 1)(s 1) c)F(s) = d)F(s) = %3D…

A:

Q: Use the superposition principle to solve Laplace's equation (1) in Section 12.5 a²u a2u = 0, 0 < x <…

A:

Q: Consider the minimization problem M(p, y) = min x -U(x) s.t. p1 · x1 + ... + pn · xn < y where U:Rn…

A:

Q: If L(x, y) is the local linear approximation of f(x, y)= x²+2y3+ 9:" at (2, 1) where g(2, 1) = 4,…

A: Given: f(x,y)=x2+2y3+g(x,y)x g(2,1)=4 gx(2,1)=4 gy(2,1)=6 At (2,1).…

Q: 5.19 Consider the mapping T: L'(0,2) L'(0, 2), (Tf)(x):= f(t)dt. (i) Show that T indeed maps L'(0,…

A: We will use the basic knowledge of functional analysis to answer both the parts of this question…

Q: Compute the weak solution to the problem (a(x)u')' – u = x _for x € (0, 1), u(0) = u(1) = 0 , 1 with…

A: For the solution follow the next steps.

Q: (a) Show that V(x, y) = ln(1 + ²) + y² is a Lyapunov function for the system x' (t)= x(y-1), y(t)=-…

A: As per your query the solution of the part (a) of the problem 4 has been given in the next step.

Q: 3. a. Assume that P(x) and Q(x) are continuous over the interval [a, b]. Use the Fundamental Theorem…

A: (a)

Q: 6. Suppose that the coefficient functions p(t) and q(t) are continuous in the interval (0, t), and…

A: if you find my answer helpful please put thumbs up. thank you. the required solution is given below…

Q: If L(x, y) is the local linear approximation of f(x, g(2, 1)= 4, g,(2, 1)=4 and g,(2, 1)=6, then L O…

A: Use formula of linear approximation

Q: Question 3 Let F (x, y, z) = zi + yj+ xk Calculate the flux of F across S, where the normal to S…

A: (3)The given surface is S: x+y+z=3 in the first quadrant. We have to find ∬SF→.dS,as we know the…

Q: 6. If u = given by x = +2, where x+y+z= 1, prove that the stationary value of u is y b a+b+c' a…

A:

Q: Given a function f(x) = f(x1, x2) = (x2 − x 2 1 )(x2 − 2x 2 1 ), where x ∈ S =  [x1, x2] T : x1 ∈…

A: Given a function fx=fx1, x2=x2-x12x2-2x13, where x∈S=x1, x2T: x1∈ℝ, x2&lt;0. (a) Find out the nature…

Q: 5.6 Exercises Problem 5.1. Consider the two-point boundary value problem u(0) = u(1) = 0. -u" = f,…

A:

Q: f(t)=t2 and g(t)=t, then the Wronskian W(f,g)(t) is: (A)-t2 (B) t-t2 (C) t (D) t2

A: Given query is to find Wronskian W(f,g)(t).

Q: 2) Aşağıdaki fonksiyonların analitik olup olmadığını Cauchy-Riemann koşulları ile cevaplayın. a)…

A: given function fz=e-xsiny+ie-xcosy to show - the given function is analytic using cauchy riemann…

Q: Theorem 5-6. (a) If X and Y are independent continuous r.v.'s, then the p.d.f. of U = X + Y is given…

A:

Q: 1. Indicate whether the following statements are true or false. No justification necessary. If g(x,…

A: As per the bartleby policy, I will be solving first three options. For other's please post it again.

Q: (ii) Show that the quadratic function f(x)=x" Ax+2b"x+c with A = AT E R²x",bER",c ER is strongly…

A: Sol

Q: Theorem 2.10 Suppose X, Y, Xn are integrable. (i) If X = a a.s., then E(X N) = a, a.s. (ii)…

A: Given X, Y, Xn are integrable

Q: 1. Determine whether the vector-valued function 't² – 9 v - F(t) = t - 3 V3t – 1, t - 3 if t < 4, t…

A: We need to check continuity at t=3.

Q: The function f(x)= V(e 1)(3 x) satisfies the hypothesis of Rolle's Theorem on 1,3 [1, 3].

A: Three conditions for a function to satisfy Rolle's hypothesis within the interval [a, b] are: The…

Q: 5. Given that G(x, y) = (x² + 1, y²) and F(u, v) = (u + v, v²), compute the Jacobian derivative…

A: