In calculus, critical points are the values where the derivative of a function is either zero or undefined. These points are essential in determining the maximum and minimum values of a function. The first derivative test is commonly used to find critical points.

To find critical points using the first derivative test, follow these steps:

1. Take the derivative of the function.
2. Set the derivative equal to zero and solve for the variable.
3. Find any critical points by evaluating the derivative where it is undefined.

Let's work through an example to understand the process better. Consider the function f(x) = 2x^3 - 9x^2 + 12x + 6. Taking the derivative, we get f'(x) = 6x^2 - 18x + 12. Setting f'(x) equal to zero, we have 6x^2 - 18x + 12 = 0. Solving this quadratic equation, we find x = 1 and x = 2 as the critical points. Evaluating f'(x) where it is undefined, we see that no additional critical points exist for this function.

Remember that critical points are significant in identifying maximum and minimum values. Further exploration of these concepts will be discussed in the next post.