Paper, Order, or Assignment Requirements
Please see the attached document for further information.
All calculation work must be typed out.
- There is a cardboard box that each measures x, y and z for its length, width Please find the combination of x, y and z when volume is at its largest when x + y + z = 270.
(Hint: the answer is when x = y = z = 90. Use the Theory of two variable functions to confirm this)
- For one variable function u(x), assume u′(x) > 0 and u′′(x) < 0 for arbitrary number x > 0.
We want to consider the following function:
We want to maximize f(x1,x2) under the following condition:
Optimum Solution will be written as x∗1, x∗2 .
Now, please prove that x∗1 < x∗2 will be always true , using first derivative of optimization. (Hint: Prove by using the property of u(x). Please also think of the shape of graph using the sign of differentiation. x∗1, x∗2 can not specifically be calculated.
Theory:Function g(z) is monotone increasing function on R (that is, for arbitrary real number z and w, if ,z < w then g(z) < g(w)).
『g ◦f(x∗, y∗) ‘s maximum value at domain/territoryD .』⇔『f(x∗, y∗)’s maximum value at domain/territory D 』
(Proof) :We assume g◦f(x,y) ≤ g◦f(x∗,y∗) is true for arbitrary (x,y) ∈ D. Here, we assume there is (x′, y′) ∈ D for f(x′,y′) > f(x∗,y∗). g increases monotonically so, g ◦ f(x′, y′) > g ◦ f(x∗, y∗). Next, we assume f(x, y) ≤ f(x∗, y∗) is true for arbitrary (x, y) ∈ D. Here, when we think about composite function g◦ f , we realize g increases monotonically so ,g ◦ f(x, y) ≤ g ◦ f(x∗, y∗).
Using this theory, we realize that the solution does not change even we use monotonic transformation for objective function.
Please solve the following problem using the above theory.
- n is a positive whole number and r is natural number r ≤ n. For function f (p, q) = pr qn−r , please find p in which maximizes the value under the condition p + q = 1.(Hint: Substitute Conditional equation q = 1 − p into objective function. Then, take logarithm and perform monotonic increase conversion before solving the problem.)
- We know the standard deviation of binomial distribution B(n, p) is √np(1 − p). Lets say that n is provided. For this standard deviation, when will p be at its maximum?