5. (a) Suppose f is any function with continuous second-order partial derivatives such that f(0,0) =...
Please do question 5a and 5b 4. In this problem we analyze the behavior of the polynomial f (x, y) = ax² + bxy + cy? (without using the Second Derivatives Test) by identifying the graph as a paraboloid. (a) By completing the square, show that if a + 0, then b 2 4ac - 62 f(x, y) = ax² + bxy + cy? = a [( 2 + Y + 2a 4a2 (b) Let D = 4ac – 62....
Suppose that f is a twice differentiable function and that its second partial derivatives are continuous. Let h(t) = f(x(t), y(t)) where x = 2e and y = 2t. Suppose that f:(2,0) = 4, fy(2,0) = 3, fx=(2,0) = 2, fyy(2,0) = 3, and fxy(2,0) = 2. Find out that when t=0.
Find the first-order partial derivatives (fr. f,) and second-order partial derivatives (fxxıfyy, fxy, fyx) of the following functions. a. f(x,y)=x’y+x’y? +x+y? b. f(x, y) = (x + y)? Find the critical points at which the following function may be optimized and determine whether at these points the function is maximized, minimized or at a saddle point. z = 5x2 – 3y2 – 30x + 7y + 4xy
Let f be a function having derivatives of all orders for all real numbers. Selected values of f and its first four derivatives are shown in the table above. (a) Write the second-degree Taylor polynomial for f about x = 0 and use it to approximate f(0.2). (b) Let g be a function such that g(x) =f(x3). Write the fifth-degree Taylor polynomial for g', the derivative of g, about x = 0. (c) Write the third-degree Taylor polynomial for f about x =...
15 4 23 Let fbe a function having derivatives of all orders for all real numbers. Selected values of fand its first four derivatives are shown in the table above. 6. a) Write the second-degree Taylor polynomial for faboutx0 and use it to approximate f(0.2). (b) Let g be a function such that g(x) -fx Write the fifth-degree Taylor polynomial for g', the derivative of g, about x = 0 We were unable to transcribe this image 15 4 23...
Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f continuous at (0,0). 2. Compute the partial derivatives of f at any (x, y) E R2. Are the partial derivatives continuous (0,0). at (0,0) (0,0) and 3. Compute the second derivatives 4. Compute the linear approzimant of f at (0,0). Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f...
Let f(x,y) = (x" + 2?y?)!. compute all second-order partial derivatives of fat (0,0), if they exist, and determien wheter dæðyəyər at (0,0).
Suppose (1,1) is a critical point of a function f with continuous second derivatives. In each case, what can you say about f? (6 Pts) (a) fxx(1,1) = −3, fxy(1,1) = 1, fyy(1,1) = 2 (b) fxx(1,1) = −4, fxy(1,1) = 2, fyy(1,1) = −1 (c) fxx(1,1) = −4, fxy(1,1) = −2, fyy(1,1) = −2 (d) fxx(1,1) = 4, fxy(1,1) = 3, fyy(1,1) = 3
72 Partial Derivatives: Problem 16 Next Previous Problem List (1 point) Suppose the f(x, y) is a smooth function and that its partial derivatives have the values, f(0,-4) 5 and f,(0, -4) =-1. Given that f(0,-4) = 0, use this information to estimate the value of f(1,-3) Note this is analogous to finding the tangent line approximation to a function of one variable. In fancy terms, it is the first Taylor approximation. Estimate of (integer value) f(1,-3) 72 Partial Derivatives:...
Let f(x) be a function on Rn with continuous first-order partial derivatives and let M be a subspace of Rn. Assume that x* ∈ M. Suppose x* minimizes f(x) on M. Prove that ∇f(x*) ∈ M⊥ Assume, in addition, that f(x) is convex and ∇f(x*) ∈ M⊥. Prove that x* is a global minimizer of f(x) on M.