As a subject matter expert in the field of calculus and mathematical analysis, I'm delighted to delve into the intricacies of critical points, particularly at a cusp. Critical points are a fundamental concept in calculus, especially when studying the behavior of functions. They are points where the derivative of a function either does not exist or is equal to zero. These points are significant because they often mark the locations of local maxima, local minima, or points of inflection. Now, let's address the question of whether there can be a critical point at a cusp. A cusp is a type of singularity where the graph of a function has a sharp turn. It's a point where the function's graph is not differentiable, meaning the derivative does not exist at that point. This is precisely the scenario where we encounter a critical point due to the non-existence of the derivative. To understand this better, let's consider a function that has a cusp. A classic example is the function \( f(x) = x^2 \sin(\frac{1}{x}) \), where \( x \) is not equal to zero. At \( x = 0 \), the function has a cusp. To analyze this, we would look at the first derivative of the function: \[ f'(x) = 2x \sin(\frac{1}{x}) - \cos(\frac{1}{x}) \] As \( x \) approaches zero, the term \( \cos(\frac{1}{x}) \) oscillates between -1 and 1, and thus does not approach a specific limit. However, the term \( 2x \sin(\frac{1}{x}) \) approaches zero because the sine function is bounded and is being multiplied by \( x \), which goes to zero. Therefore, the behavior of \( f'(x) \) as \( x \) approaches zero is not defined, and we say that \( f'(0) \) is undefined. This undefined behavior at \( x = 0 \) signifies a critical point. However, it's important to note that this critical point does not necessarily correspond to a local extremum. To determine if it's a local maximum or minimum, we would typically use the first or second derivative tests. But at a cusp, these tests are not applicable because the derivative does not exist. In the context of local extrema, critical points where the derivative is undefined can occur at corners or at cusps. It's also possible for the derivative to be undefined due to even worse behavior, such as a discontinuity in the derivative itself, but such cases are less common and more complex to analyze. In summary, yes, there can be a critical point at a cusp. It is characterized by the non-existence of the derivative at that point, and while it is a critical point, it does not follow the usual rules for determining local extrema due to the lack of differentiability. read more >>

The next type of critical point is that where f '(x) is undefined. In the context of local extrema, this can happen at a corner or at a "cusp" as shown at the right. (It can also happen that f '(x) is undefined due to worse behavior, but we will not encounter this very much.)read more >>