what is the second derivative test?

Question

what is the second derivative test? how can you test for concavity? how can you find relative extrema using second derivative test? how can you find points of inflection? how do you know when a function increases or decreases?

Answer 1

the second derivative test is to test if the curve/path is concaved upward or downward.

aside from this, you could also know if the function's extremum (extremum from the first derivative equated to zero) is a maximum or minimum.
How?

The extrema is through the first derivative test equating it to zero and then solving it.

To know if it's a maximum, get the second derivative, if the answer is negative, it is a maximum. If it is positive, it is a minimum.

Answer 2

f `(x) = 0 gives turning points
f "(x) gives nature of turning points
f "(x) +ve gives a MIN. turning point at x
f "(x) - ve gives a MAX turning point at x
f "(x) = 0 gives a point of inflection at x