Final answer:
Critical points occur where the derivative of a function is zero or undefined. For functions with an absolute value, the critical points often coincide with where the function is not differentiable, typically at the corner of the graph. The function f(x) = |x - 2| + 3 has one critical point at x = 2.
Step-by-step explanation:
To find the critical points of a function involving an absolute value, we need to first understand what a critical point is. A critical point of a function occurs where the derivative is either zero or undefined. An absolute value function typically has a corner or cusp where the derivative is undefined, which can be a critical point.
Let's consider a function f(x) that includes an absolute value, such as f(x) = |x - 2| + 3. To find the critical points, we differentiate the function where it is differentiable, and consider the case where the function is not differentiable, like at the sharp corner of the absolute value graph.
Step 1: Identify where the function is not differentiable
In the case of f(x) = |x - 2| + 3, the function is not differentiable at x = 2, since this is where the graph of the absolute value function makes a sharp turn. Therefore, x = 2 is one of the critical points.
Step 2: Find where the derivative is zero
For x values less than 2, f(x) can be rewritten as f(x) = -(x - 2) + 3, and the derivative f'(x) is -1 which is never zero. For x values greater than 2, f(x) can be rewritten as f(x) = (x - 2) + 3, and the derivative f'(x) is 1 which is also never zero. Therefore, there are no critical points for f(x) from the derivative being zero.
In summary, the only critical point for this function is at x = 2.