Question
Consider the following. f(x,y)=x^2 +4xy +2y^4 Find all points (x, y) where f(x, y) has a possible relative maximum or minimum. Then use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points. If the second-derivative test is inconclusive, so state. So far, I got (df/dx)2x+4y, (d2f/dx2)2, (df/dy)4x+8y^3, (d2f/dy2)24y^2, (df/dxy)4 I am supposed to have 3 answers for x, but I do not see how... I do not need help doing the second derivative test by the way.
Explanation / Answer
f ' (x) = x2 - 6x + 5 = (x-1)(x-5) = 0 when x = 1, 5. Now f '' (x) = 2x - 6, so f '' (1) = -4 < 0 implies f has a local max at x = 1; f '' (5) = 4 > 0 implies f has a local min at x = 5