Final answer:
The points on the curve where the tangent is horizontal can be found by setting the derivative, \(f'(x)\), equal to zero.
Step-by-step explanation:
To find points on the curve where the tangent is horizontal, we need to locate the values of \(x\) where the derivative, \(f'(x)\), is equal to zero. A horizontal tangent occurs when the slope of the curve is zero at a particular point. Mathematically, this is represented by setting the first derivative, \(f'(x)\), equal to zero and solving for \(x\).
Next, by using the second derivative test, we can determine whether these critical points correspond to minima, maxima, or points of inflection. If the second derivative, \(f''(x)\), is positive at a critical point, it indicates a local minimum, while a negative second derivative suggests a local maximum. If \(f''(x)\) is equal to zero, the test is inconclusive, and further analysis may be needed.
Additionally, points where \(f(x) = 0\) can be identified, but this may not necessarily correspond to horizontal tangents. Finally, the expression \(dy/dx\) is another notation for the first derivative \(f'(x)\), so setting \(dy/dx = 0\) is equivalent to finding points where the tangent is horizontal.
In summary, to locate points on the curve where the tangent is horizontal, the primary approach involves setting the first derivative, \(f'(x)\), equal to zero and examining the second derivative, \(f''(x)\), to determine the nature of these points.