Second derivative test in Wikipedia says that:
For a real function of one variable:
If the function f is twice differentiable at a stationary point x, meaning that $\ f^{\prime}(x) = 0$ , then:
If $ f^{\prime\prime}(x) < 0$ then $f$ has a local maximum at $x$. If $f^{\prime\prime}(x) > 0$ then $f$ has a local minimum at $x$. If $f^{\prime\prime}(x) = 0$, the second derivative test says nothing about the point $x$, a possible inflection point.
For a function of more than one variable:
Assuming that all second order partial derivatives of $f$ are continuous on a neighbourhood of a stationary point $x$, then:
if the eigenvalues of the Hessian at $x$ are all positive, then $x$ is a local minimum. If the eigenvalues are all negative, then $x$ is a local maximum, and if some are positive and some negative, then the point is a saddle point. If the Hessian matrix is singular, then the second derivative test is inconclusive.
My question is why in the multivariate case, the test requires the second order partial derivatives of $f$ to be continuous on a neighbourhood of $x$, while in the single variable case, it does not need the second derivative to be continuous around $x$? Do both also require that the first derivative and the function itself to be continuous around $x$?
- Similarly, does first derivative test for $f$ at $x$ need $f$ to be continuous and differentiable in a neighbourhood of $x$?
- For higher order derivative test, it doesn't mention if $f$ is required to be continuous and differentiable in a neighbourhood of some point $c$ up to some order. So does it only need that $f$ is differentiable at $c$ up to order $n$?
Thanks for clarification!