1) a) Given
Now,
Since these are all positive, the quadratic form induced by A is positive definite.
b) Given
A matrix B is negative definite if for odd and for even.
Since <0 and >0, we have B is a negative definite matrix.
c)Given
Now,
Neither of the condition of positive nor negative definite holds and determinant not equal to zero, C is indefinite.
d) Given
Now,
Hence D is positive definite.
e) Given
Now,
E is Indefinite.
2. Given
Quadratic form is
thus the form is positive definite.
3. a)
Set first derivative equal to zero to get the critical points
thus, x=0, y=0. (0, 0) will be our critical point.
and
Therefore is a local minimum.
b)
Set first derivative equal to zero to get the critical points
thus, x=3, y=1. (3, 1) will be our critical point.
and
Therefore is local maximum.
c)
Set first derivative equal to zero to get the critical points
will be our critical point.
. Then is not a local maximum nor a local minimum
d)
Set first derivative equal to zero to get the critical points
(0, 0) will be our critical point
f(0, 0)=0 is not a local maximum nor a local minimum.
e)
Set first derivative equal to zero to get the critical points
(0,0) will be our critical point.
(0, 0) will be a saddle point.
ECON 1111A/B Mathematical Methods in Economics II 2nd term, 2018-2019 Assignment 6 Show your step...
9 Exercise 1. For the following functions, which you previously analyzed for Homework 16, calculate the Hessian matrix at the points that you previously identified to be local extrema. Then classify whether these extrema are local minima or local maxima. (a) f(x, y) = –2x2 - y² + 2x (b) f(x,y) = -xy – 2y2 (c) —2+y2 f(x, y) = ce (d) f(x, y) = xyey (e) f(x, y) = x² + 4xy + y2 + y (f) f(x, y)...
2. For each function, find all critical points and use the Hessian to determine whether they are local maxima, minima, or saddle points. (a) f(x,y,z) = x — 2 sin x – 3yz (b) g(x, y, z) = cosh x + 4yz – 2y2 – 24 (c) u(x, y, z) = (x – z)4 – x2 + y2 + 6x2 – 22
Calculus III 1) Identify each of the following surfaces: а) z' %3x? - 5у" b) z 4x2-4 y c) z2+3x2-5y = 4 d) z2x23-5y e) у3х* 2) Find and classify all of the critical points for f(x, y)=xy -x2 + y'. 3) Find the maximum and minimum values of f(x,y)=xy over the ellipse х* + 2y %3D1. 4) Let fx, y) x3 -cosy a) Find the first order Taylor polynomial for f(x,y) based at (1,7). b) Find the sccond order...