The Importance of Theorem 3.2: Understanding Critical Numbers and Relative Extrema in Mathematics

Theorem 3.2 – Relative Extrema Occur Only at Critical Numbers (3.1)

Theorem 3

Theorem 3.2 states that relative extrema of a function can only occur at critical numbers. To understand this theorem, let’s first review what critical numbers and relative extrema are.

A critical number of a function is a point in the domain where either the derivative does not exist or is equal to zero. In other words, it is the x-value at which the slope of the function is either undefined or equal to zero.

On the other hand, relative extrema are points on the function where it reaches a local maximum or minimum. A relative maximum occurs when the function transitions from increasing to decreasing, and a relative minimum occurs when the function transitions from decreasing to increasing.

The significance of Theorem 3.2 is that it tells us that if a function has a relative maximum or minimum, it can only occur at a critical number. This means that if we want to find the relative extrema of a function, we only need to look at the critical numbers.

To understand why this is the case, let’s consider the behavior of the function on both sides of a critical number. If the function is increasing on one side of the critical number and decreasing on the other, then it must reach a relative maximum or minimum at the critical number.

If the function is increasing on both sides or decreasing on both sides of the critical number, then it cannot have a relative maximum or minimum at that point. This is because for there to be a relative maximum or minimum, there must be a change in the slope of the function, which is indicated by the derivative being zero or undefined.

In summary, Theorem 3.2 states that if a function has a relative maximum or minimum, it can only occur at a critical number. This theorem helps us focus our attention on critical numbers when looking for relative extrema, simplifying the process of finding and identifying these important points on a function.

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