Let f R R be defined by f(x) 2x4 x4 sin(1 x) for x 0 and f(0) 0 Show that f has an absolute minimum at x 0, but that its derivative has both positive and negative values in every neighborhood of 0

Question: Let f : R R be defined by f(x) := 2x4 + x4 sin(1/x) for x 0 and f(0) := 0. Show that

Let f : R → R be defined by f(x) := 2x4 + x4 sin(1/x) for x ≠ 0 and f(0) := 0. Show that f has an absolute minimum at x ¼ 0, but that its derivative has both positive and negative values in every neighborhood of 0.

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