A jewelry floor safe with a square base is to be made so as to have a total volume of 648 cubic inches. The side material for the safe costs $1 per square inch. The material for the top costs $4 per square inch while the material for the bottom costs $2 per square inch. What, in inches, would be the height of the most economical box which could be made given these conditions?

12 inches
15 inches
18 inches
21 inches
24 inches
27 inches
none of these

A hotel manager has 200 rooms available for rent each night. Experience shows that if she charges $40 per night, she will have full occupancy, but that for each $5 that she increases the price, 4 rooms become vacant. All other factors being equal, what will her maximum revenue be?

$16,820
$13,520
$8820
$12,000
$11,520
$18,000
none of these

Solution Summary

The maximum values of functions are found. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Rules and Applications of the Derivative
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1. Use the Product Rule to find the derivatives of the following functions:
a. f(X) = (1- X^2)*(1+100X)
b. f(X) = (5X + X^-1)*(3X + X^2)
c. f(X) = (X^.5)*(1-X)
d. f(X) = (X^3 + X^4)*(30

Please see attached file for full problem description.
The revenue derived from the production of x units of a particular commodity is million dollars. What level of production results in maximum revenue? What is the maximum revenue?
a. a. Maximum at x = 8 and maximum revenue is R(8) = 32 (million dollars)
b. b. Maxi

#4
What happens to concavity when functions are added?
a) If f(x) and g(x) are concave up for all x, is f(x) + g(x) concave up for all x?
Yes
b) If f(x) is concave up for all x and g(x) is concave down for all x, what can you say about the concavity of f(x) + g(x)? For example, what happens if f(x) and g(x) are both polyn

Please see the attached file - Don't mind the graphing portion, work what can be worked. I have to input answers not write out the whole equation.
7. Find the vertex, line of symmetry, and the maximum or minimum value of a quadratic function, and graph the function.f(x) = -x +6x +2
The vertex is
8. Find the vertex,lin

For each of the following functions, find the maximum and minimum values of the function on the circular disk: x2 + y2 [less than or equal to] 1. Do this by looking at the level curves and gradients.
*(For functions please see attachment)

A Monopolist's Demand and Total Cost functions are:
P= 1624 -4Q
TC= 22,000 + 24Q -4Q(squared) + 1/3Q (to the third power)
Where Q is output produced and sold
a. At what level of output and sales (Q) and price (P) will Total Profits be maximized?
b. At What level of output and sales (Q) and price (P) will Total Rev

Please help with the following mathematics problems.
(a) Let f be a differentiable functions defined on an open set U. Suppose that P is a point in U that f(P) is a maximum, i.e.
f(P) >= f(X) for all X E U
Show that grad f(P) =0
(b) Find the global maximum of the function
f(x,y)=x^3 +xy
defined on the set
S={(x,y)|-1<=

(d) Does the conclusion of the Maximum-Minimum Theorem always hold for a bounded function f : R --> R that is continuous on R? Prove or give a counterexample.
(a) Fix a, b E R, a < b. Prove that if f [a, b] -->R is continuous on [a, b] and f(x)≠0 for all x E [a, b], then 1/f(x) is bounded on [a, b].
(b) Find a, b E R, a