The Second Derivative test

Let's say we have a function $f(x)$ and we want to know at what value of $x$ it has a local maximum.

For this we can use following summary:

  1. If $f'(x) = 0$ and $f''(x) < 0$, then the function $f()$ has a maximum at $x$
  2. If $f'(x) = 0$ and $f''(x) > 0$, then the function $f()$ has a minimum at $x$
  3. If $f'(x) = 0$ and $f''(x) = 0$, then we gain no extra information (about the original function having a maximum or a mnimum)

The second derivative

You can use the black dots to change the shape of the second derivative.

The first derivative

The derivative of a constant is zero. So the above second derivative determines the shape of the first derivative up to a constant
You can use the black dot to alter this constant.

The function

The derivative of a constant is zero. So the above second derivative determines the shape of the first derivative up to a constant
You can use the black dot to alter this constant.