Extreme times in financial markets

The highest and lowest values of a function, relative to a reference axis, are known as extreme values. These include absolute maximum and absolute minimum values, which represent the highest and lowest points the function reaches across its entire domain. Within a restricted portion of the function, the highest and lowest values are referred to as local maximum and local minimum values, respectively.Periodic functions, such as sine and cosine, show extreme values at infinitely many points due...

Understanding the maximum and minimum values of a function is essential for analyzing its overall behavior. These values, often referred to as extrema, provide insight into how a function behaves across its domain. In mathematical terms, extrema can be either local—representing peaks and valleys within a limited region—or absolute, indicating the highest or lowest points over an entire interval.A function’s extrema occur at critical numbers, which are values in the domain where the derivative...

Imagine an asset price that crashes to a low point, rebounds sharply as bargain-hunters step in, and then gradually declines. Such behavior can be modeled with a smooth function whose turning points represent locally overvalued and undervalued regions. A convenient example that captures rebound followed by decay is:The high and low points of this curve are identified using the first derivative test, which determines where the function changes from increasing to decreasing or vice versa. To...

Certain mathematical functions exhibit unpredictable or highly variable behavior near specific input values, making direct evaluation of their limits challenging. This complexity may arise from rapid oscillations or irregular patterns that obscure the function’s trend. In such cases, the Squeeze Theorem offers a reliable method for determining limits.According to the Squeeze Theorem, if a function is confined between two other functions near a particular point, and both outer functions approach...

The function that decreases as the input becomes very large provides a clear example of how mathematical functions can behave at extreme values. When the input increases continuously, the output becomes smaller and smaller, getting closer to a particular fixed value. Although the output never actually reaches this value, it moves nearer to it without limit. This behavior is a fundamental concept in understanding how functions behave as the input grows indefinitely. The graphical representation...

When observing how a curve behaves near a specific point along the horizontal axis, there are cases where the curve’s height increases or decreases without limit as the position draws closer to that point. The curve does not settle at any particular value; instead, the values grow more extreme—upward or downward—the nearer they get. No defined value exists exactly at that location, yet the surrounding behavior becomes more dramatic, indicating a sharp change in direction.The values may rise...