To find local extrema, which means the highest and lowest points of a function, we can use a method called the Second Derivative Test. This method helps us understand how the function curves at certain points.

Here’s a simple way to do it:

1. First Step - Find Critical Points:

   • Begin by finding the critical points of the function.
   • You need the first derivative of the function, written as $f'(x)$.
   • Set $f'(x) = 0$. This will give you the points where the function might have a local maximum (highest) or minimum (lowest).

2. Second Step - Calculate the Second Derivative:

   • After finding the critical points, calculate the second derivative, written as $f''(x)$.
   • This tells us about the curvature of the function at those points.

3. Third Step - Use the Second Derivative Test:

   • If $f''(c) > 0$: The graph is curving upwards at point $c$, which means $c$ is a local minimum (like the bottom of a "U").
   • If $f''(c) < 0$: The graph is curving downwards at point $c$, which means $c$ is a local maximum (like the top of an "n").
   • If $f''(c) = 0$: The test doesn’t give a clear answer. You may need to try the First Derivative Test for more information.

Example: Let’s look at the function $f(x) = x^3 - 3x^2 + 4$.

• First, we find the first derivative: $f'(x) = 3x^2 - 6x$ Set this to zero: $3x^2 - 6x = 0$ This means we can factor it: $x(3x - 6) = 0$ So, $x = 0$ or $x = 2$ are our critical points.

• Next, we find the second derivative: $f''(x) = 6x - 6$

• Now, we look at the critical points:

  • For $x = 0$: $f''(0) = 6(0) - 6 = -6 \ \text{(local maximum)}$
  • For $x = 2$: $f''(2) = 6(2) - 6 = 6 \ \text{(local minimum)}$

By using the second derivative test, you can easily find the local maximum and minimum points of a function and see how it curves at those points!