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- 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?