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- The function f(x,y) = R x y cos(t2)dt has exactly one critical point in the domain 0 < x < 2, 0 < y < 2. Find this critical point and determine whether it gives a relative maximum, minimum or saddle point for this funciton.Prove that the function for (x,y) ∈ R2, has a local maximum, a local minimum and a saddle point.The paraboloid z = x2 + y2 - 4x + 2y + 5 has a local minimum at (2, -1). Verify the conclusion of as shown for this function.
- Let f(x, y) = cos(x)cos(y). Find all critical points of f which lie in the square {(x, y) ∈ R2 : −1 < x < 4 and − 1 < y < 4} and classify each as a local maximum, local minimum, or saddle point.The function f(x,y,w)=√6xy+5w^2-x^2-0.5y^2-4y+7 has a unique stationary (i.e., critical) point. Find the characteristic roots of the Hessian of this function, order them from the smallest to the largest. r1=? r2=? r3=? And, using the characteristic roots test, determine for each whether it is a local (global) maximum, a local (global) minimum, or neither.Determine the absolute maximum and minimum values of F over the set D, which consists of the closed tirangular region in the xy plane with vertices (0,0), (0,6) and (6,0).
- Although they are not defined on all of space R3, the fields associated with Exercises 18–22 are conservative. Find a potential function for each field and evaluate the integrals as in Example 6.3 Show that the square integrable function f(x) = sin( πk log x/ log 2 )for k ≥ 1 are orthogonal over the interval 1 ≤ x ≤ 2 with respect to the weight function r(x) = 1/ x . Obtain the norms of the functions and construct the othornormal set.Let f(x, y) = 3y2−2y3−3x2+6xy. Find all critical points of f and use the Second Derivatives Test to determine whether f has a saddle point or a relative maximum or minimum at each of those points. Show complete solution.
- Consider the function f(x, y) = x2 + 3y2 + 2y on the closed disk S: x2 + y2 < 1. a) Find the critical points in the interior of S using the first and second derivative tests, and decide if they are local maxima, minima, or neither. b) Find the maximum and minimum of f(x, y) on the boundary of S by using Lagrange multipliers on the boundary of S. (i.e. optimize f(x, y) subject tothe constraint x2 + y2 = 1).(c) Using the results of (a) and (b), conclude what the global maximum and minimum values of f(x, y) are, and where they are attained.Label the following statements as true or false.{a) The set of solut ions to an nth-order homogeneous linear differentialequation with constant coefficients is an n-dimensional subspace ofcoo. (b) The solution space of a homogeneous linear differential equationwith constant coefficients is the null space of a differential operator.(c) The auxiliary polynomial of a homogeneous linear differentialequation with constant coefficients is a solution to the differentialequation.(d) Any solution to a homogeneous linear different ial equation withconst ant coefficients is of the form aect or atkect , where a and care complex numbers and k is a positive integer. (e) Any linear combination of solutions to a given homogeneous lineardifferential equation with constant coefficients is also a solution tothe given equation.Determine Taylor's formula of order 1 and center a = (0, 0) of f(x)=ex1sin(x2).