Intuitive Explanation
The idea behind the first derivative test is to examine the monotonic properties of a function just to the left and right of a given point in its domain. If the function "switches" from increasing to decreasing at the point, then close to that point, it will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then close to that point, it will achieve a least value at that point. If the function fails to "switch", and remains increasing or remains decreasing, then no highest or least value is achieved.
The general idea of examining monotonicity does not depend on calculus. However, calculus is introduced because there are sufficient conditions that guarantee the monotonicity properties above, and these conditions apply to the vast majority of functions one would encounter.
Read more about this topic: First Derivative Test
Famous quotes containing the words intuitive and/or explanation:
“Ezra Pound still lives in a village and his world is a kind of village and people keep explaining things when they live in a village.... I have come not to mind if certain people live in villages and some of my friends still appear to live in villages and a village can be cozy as well as intuitive but must one really keep perpetually explaining and elucidating?”
—Gertrude Stein (1874–1946)
“Herein is the explanation of the analogies, which exist in all the arts. They are the re-appearance of one mind, working in many materials to many temporary ends. Raphael paints wisdom, Handel sings it, Phidias carves it, Shakspeare writes it, Wren builds it, Columbus sails it, Luther preaches it, Washington arms it, Watt mechanizes it. Painting was called “silent poetry,” and poetry “speaking painting.” The laws of each art are convertible into the laws of every other.”
—Ralph Waldo Emerson (1803–1882)