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Power Mean Inequalities

This chain of inequalities forms the foundation for many other classical inequalities. See how the four common "means" - arithmetic, geometric, harmonic, and quadratic - relate to each other.

Problem Solving

         

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.)

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