Math Help

First and Second Derivative Tests

In a Nut Shell: Once the critical numbers for a function, f(x), are found the first and

second derivative tests provide information on local maxima and local minima of the

function.

The First Derivative Test

Let c be a critical number of a continuous function, f(x). Then

If df/dx changes from positive to negative at c, then f(x) has a local maximum at c.

If fdf/dx changes from negative to positive at c, then f(x) has a local minimum at c.

If df/dx does not change sign (either both positive or both negative at c), then

f(x) has no local maximum or local minimum at c.

The Second Derivative Test

Let c be a critical number of a continuous function, f(x) and let d²f/dx² be continuous

near c.

If df/dx = 0 at c and d²f/dx² > 0, then f(x) has a local minimum at c.

If df/dx = 0 at c and d²f/dx² < 0, then f(x) has a local maximum at c.