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If f: R \to R is a differentiable function, x_0 is a point in R such that f'(x_0) = 0, and f''(x_0) > 0 (so in particular f'' exists at x_0), then

If f: R to R is a differentiable function, x_0 is a point in R such that f'(x_0) = 0, and f''(x_0) > 0 (so in particular f'' exists at x_0), then there is a d > 0 so that f(x_0) < f(x) for all x in (x_0 - d, x_0 + d) that aren't equal to x_0.

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You may probably know the Taylor formula for C 2 functions, I am stating
here.
Taylor formula for C 2 function:
Suppose f is C 2 (twice differentiable) function, then for any x0 we have
f (x0 + h)...

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