Why the peaks and valleys of a curve must be flat
In short: If you are looking for the highest or lowest point on a smooth curve, you are searching for a moment of perfect balance. Fermat's Theorem provides the mathematical proof for why these peaks and valleys occur exactly where the rate of change hits zero.
Fermat's Theorem is a fundamental principle in Calculus 1 that links the concept of relative extrema—the local maximums and minimums of a function—to the derivative. It establishes that if a function has a relative extremum at a specific point and the derivative exists at that point, then that point must be a critical point where the derivative equals zero.
The logic relies on the behavior of the function as you move slightly to the left or right of a peak. For a relative maximum at point c, the function value at c must be greater than or equal to the values of nearby points. By analyzing the difference between the function at c and a nearby point c plus h, we can see that the slope must be non-positive when approaching from the right and non-negative when approaching from the left.
Mathematically, this creates a squeeze. If the limit of the difference quotient (the derivative) exists, it must satisfy both conditions simultaneously. Because the derivative must be both less than or equal to zero and greater than or equal to zero at that specific point, the only possible value for the derivative is exactly zero. This confirms that at the very top of a hill or the bottom of a valley, the instantaneous rate of change momentarily vanishes.
This theorem serves as a primary tool for optimization. By finding where the derivative is zero, mathematicians can systematically identify potential candidates for the best or worst possible outcomes in various models, from physics to economics.
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Sources
- Summary by Intellect · source: Fermat's Theorem - Application of Derivatives - Calculus 1
- Also: Calculus I - Proofs of Derivative Applications Facts
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