Final answer:
Option b), mentioning points where the second derivative is zero, indicates points of inflection rather than horizontal tangents. Option c) refers to vertical tangents or cusps, and option d) refers to places where the function has discontinuities, which is different from the concept of a horizontal tangent.
Step-by-step explanation:
The points of horizontal tangent on a curve are found at locations where the derivative of the function, which represents the slope of the tangent line, is equal to zero. This is because a horizontal tangent line has a slope of zero, indicating that there is no change in the y-value as the x-value changes at that point on the curve.
To find these points, one must first calculate the first derivative of the function and then solve for the points where this first derivative is equal to zero. It's important to note that a horizontal tangent might exist at a point even if a function is not differentiable there; in such cases, the function would not have a derivative at that point. However, generally, we are looking for points where the derivative exists and is zero.
Option b), mentioning points where the second derivative is zero, indicates points of inflection rather than horizontal tangents. Option c) refers to vertical tangents or cusps, and option d) refers to places where the function has discontinuities, which is different from the concept of a horizontal tangent.