So, we have been learning optimization recently. When there is a single variable function, then by Taylor series, we can write this out as
$f(x)=f(z)+f'(z)(x-z)+\frac{f''(z)}{2}*(x-z)^2+o(x^2)$
While the second-order term is positive, we locate the local min for this function. I know this is true because when the second-order term is positive, we have a concave-up-looking function that guarantees the value to be a local min. My question is, is there another way to think about this question without involving any geometric thinking? Without knowing concavity, I have no idea why for local min, the second-order term has to be positive.