Q1
a) False
This is so because, if f(x) = x, then y=x at every particular point and thus, the condition f(x) ≤ f(y) does not hold.
b)true
a function is said to have the convex nature if it is continuous over some certain interval say (a,b).
c)true
a function with a concave shape will most likely have a maxima. Only functions with a convex shape will have a minimizer instead.
d)false
this is so because, a convex function does not have argmax. It has a minimum set of values instead.
e)true
a monotonic function is either increasing or decreasing over some interval. Thus, for a monotone transformation of a function that has a maximizer, the transform will still have a maximizer too.
Q2
a)
Since the equation of x12 + x22 =1 is a curve running from 0-1, the it would be an intelligent guess that the solution we’re looking for lies at x1 = 0.5, and x2 =0.5
Maximize ln 0.5 + ln 0.5 = -1.3862
b)
Maximize x1 + x22 s.t x12 + x22 = 1
=f(x) – λ(g(x) –b)
= x1 + x22 –λ (x12 + x22-1)
= (x1 – x12λ) + (x22 – x22λ) + λ
dL/dx1 = 1-2x1λ = 0
thus, x1 = 1/2λ
DL/dx2 = x2(2-2λ) = 0
Thus, since x2 cannot be equal to zero, then
2-2λ = 0 implying that λ=1
Therefore x1 = 1/2 and x22 = 1-1/4 = ¾. So x2 = 0.866
We can therefore conclude that the maximized value is ½ + 0.8662 =1.25.
Q3
a)
L=u(x)-λ(px-I)
DL/dx=-λp
Dl/Di=-λI
Since both are equivalent to zero, the we can conclude that;
X(λp,λ I)=(P,X)
b)
from the information provided,,
u(x)=u(p,I)
y(x)=Nε(p,I)
therefore for all (p,I) and ε greater than zero, it can be clearly seen that,
px=p(p,I)=I
c)
since u(x) is strictly concave, then this implies that it has a maxima. Since x(p,I), then u(x)=u(p,I).
This in turn proves that the solution to the maximization problem, which must be concave, is a singleton; having only one single point for a solution.
academic writing services, professional academic writing services, academic writing companies, best academic writing services, online academic writing services, academic essay help, academic essay writing service, write academic papers for pay, professional academic writers, academic writing services online, custom academic writing services, best academic writing companies, online academic writing service, best academic writing, academic writing experts, academic writing help centre, academic writing help services, term paper assignment, academic writing help online