Intuitively, I would expect that the second derivative of f(x)=x^2n evaluated at x=0 would be positive for all finite values of n. This intuition is based on the interpretation of the second derivative as the rate of change of the first derivative. So, any function with a non-constant first derivative should have a non-zero second derivative. Furthermore, if the first derivative is monotonically increasing with increasing x, the second derivative should then be positive for all x.

Taking f(x)=x^4 as a specific example, the first derivative f'(x)=4x^3 is monotonically increasing with increasing x. Therefore, should not the second derivative be positive for all x, including x=0?

Taking this a step further, my mathematical intuition (such as it is) tells me that f"(0) should equal zero only in the limit of n approaching infinity. As n-> infinity, f approaches the function

f =  +infinity ( for :x: > 1 )
  =  1         ( for :x: = 1 )
  =  0         ( for :x: < 1 )

Since this function is constant in the interval (-1,1), it clearly has a constant (zero) derivative, and hence a zero second derivative as well, in the entire interval (-1,1).

In a nutshell, I would expect that f"(0) > 0 for f(x)=x^2n, n=1,2,3,.... I would furthermore expect that the value of f"(0) decreases (but remains positive) as n increases, reaching the
value of zero only in the limit n -> infinity.

Can someone tell me what is wrong with my reasoning?

Thanks in advance for your insights.

"Non-zero on an interval" and "non-zero at a point" are completely different conditions.

All the best,
Alexander Bogomolny