Maths Problems With Solution 2022

SOLUTIONS
EXERCISE ONE
(a)
In this case g: R² R, we shall only look for stationary points
∆g (x, y) =0 dg/dx(x,y)=4x+2y=0
dg/dy(x,y)=2y+2x=0
We can divide by 2, moreover the difference in (i) and (ii) yields x=0
Our system has two equations
(i) 2x+y=0
X=0 (0,0) (-1,2)
(ii) X + y=0
X=0 ( 0, 0) (-1,1)
(b)
Hessian matrix Hg(x,y)
2 1
1 1
G is a convex function has two different systems.
(c)
(0, 0) saddle point
(-1, 2) Local Maximum
(-1, 1) Local minimum

EXERCISE TWO
(a)
The Lagrangian function is then given by
L (α, x) =2x²+2xy+y²+ α (1-x-y)
Setting ∆L=0 means
L (1-x-y) =0 (i)
4x+2y- α =0 (ii)
2x+2y- α =0 (iii)
The difference (ii) – (iii) yields
(4x-2x) + (2y-2y) +(- α -- α) =0
That is 2x=0, x=0
(i) 1-x-y=0
1-y=0
Given that x=0 and y=1 then; P (0, 1,1)
(b)
Insert α =1
L (1, x) =2x²+2xy+y²+1-x-y
4x+2y-1
2x+2y-1