Recent content by daverusin

As others have noted, if the four sides of that quadrilateral have lengths a, x, b, and the diameter 2x, then 3x^2 = a^2+ab+b^2. As to the question of how the heck they came up with the 11, 2, and 7: we can scale the problem so that the diameter x is 1, and then seek *rational* values of a...

The previous response is quite correct. However, there is a "natural" continuous extension of the factorial function -- a way to "connect the dots", if you like. Look up the "Bohr–Mollerup theorem" and learn about the Gamma function. For integers x we have x! = Gamma(x+1), so perhaps you...

I stand corrected: jostpuur's example is indeed differentiable at the "nasty" spot; indeed, the simple inequality || f(x,y) - L(x,y) || <= ||v||^2 holds for all displacements v from the point (0,1). The multiple branches of the level curve through that point all leave the point in the same...

Here is a smooth function with only one critical point (at the origin), which is a local minimum but not a global minimum: f(x,y) = x^2 + y^2(1-x)^3. If you start filling the graph with water, the origin becomes the bottom of a roughly-triangular lake lying to the west of the line x=1. By the...

I'm afraid the proposed solution does not really work: this function is not actually differentiable. I tried to figure out how on Earth it could be working, by sketching the level curves. I reorganized the function a bit, putting the questionable point at the origin and trying to simplify: for...