Algebra
Power Mean Inequalities
What is the minimum possible perimeter of a rectangle with area 25?
If \( x^2 + y^2 = 1, \) find the maximum value of \( (x + y)^2. \)
Suppose that a company produces right-cylindrical soup cans with radius \( r \) and \( h \). If the volume is constrained to be a fixed value, which of the following dimensions minimizes the material per can?
Suppose \( g(x) \) is positive for all \( x \). Determine the minimum value of
\[ f(x) = \frac{16 (g(x))^2 + 1}{g(x)}. \]
What is the minimum possible surface area of an open box (that is, a rectangular prism without its top) whose volume is \( 108 \)? (Count only the exterior surface area.)