Function Optimization & Application of the Envelope Theorem

See the attached file as equations are contained within the Word file.

Consider the following function:
(see file)
where a > 0 is a parameter.

1. Find the first order condition for a critical point of this function.
2. Is this a maximum or a minimum or an inflection point?
3. Solve for x* (a), the maximizer of the function f (x; a) :Also find y* (a), the maximized value of y as a function of a
4. Find (see file) and (see file)
5. Now use the FOC from 1 and the implicit function theorem to find
6. Use the envelope theorem to find . Why does this theorem allow you to simplify your calculations with respect to point 4?

The solution uses Microsoft Equation to display properly written formulas, so please refer to the attached Word file for complete solution. The text below is contained within the file:

Consider the following function:

where a > 0 is a parameter.

1. Find the first order ...

Solution Summary

This solution shows the steps required in order to optimize a specific function. This function is parameterized and we show how to find the maximizer - optimal solution in terms of the parameter. Finally, implicit function theorem and the envelope theorem are used to double-check the answers.

While a specific function is used, a diligent student should be able to follow the steps and apply them to a similar function. Of course, this is not a substitute for a textbook and the theory behind each step must be reviewed using class material.

Consider the problem of maximizing u(c,l) subject to pc + wl = wT + Y, where c is consumption, l is leisure time, T is the total time endowment, and Y is non-wage income. Show that if leisure is an inferior good, then the labor supply function is upward-sloping.
b) Given the problem of maximizing ln x subject to α Ͱ

Use the Pythagorean Theorem to determine if an angle is the right angle (3-4-5 triangle).
What would you use the Pythagorean Theorem for? List one example either from work or personal life or any other application.

Bayes theorem is very important, since it forms the basis of much of our thinking in conditional probability and making "predictions" statistically based on historical data. In plain English, describe an application of Bayes theorem that you think will help explain it to other students.

Determine whether Rolle's Theorem can be applied to f on the closed interval [a,b]. If Rolle's theorem can be applied, find all values of c in the open interval (a,b) such that f'(c) = 0.
f(x) = sin x, [0, 2pi]

1. Calculate the ambiguity function of a signal with an envelope u(t) = Bexp(-t^2)T^2). What should be the value of B that will make the signal of the unit energy?
2. Calculate the ambiguity of a signal with a complex envelope u(t) = Bexp(-t^2/T^2)exp(jpikt^2). Note that this is the same signal as above, except for the additi

2. verify that thefunction f(x) = e^-2x satisfies the hypothesis of mean value theorem on the interval [0, 3] and find all the number c that satisfy the conclusion of the mean value theorem.
3. show that a polynomial of degree two has at most two real roots.

(See attached file for full problem description)
Let a sequence xn be defined inductively by . Suppose that as and . Show that
.
(Note that " " refers to "little oh")
HINT: Use the Mean-Value Theorem and assume that F is a continuously differentiable function.

1. Using the intermediate Value theorem, show that thefunction f has a zero between a and b.
f(x)=x^3+3x^2-9x-13, a=-5, b=-4
2. For thefunction h(x)=5x/((x+6)(x-4)), solve the following equation
h(x)=0