4.2. First Derivative Test

🔹 Definition: • Maximum: A point where a function attains the highest value locally. • Minimum: A point where a function reaches the lowest value locally.

In calculus, understanding the concepts of maxima and minima is crucial for optimization problems. A maximum point is where the function reaches its highest point within a certain neighborhood, while a minimum point is where it hits its lowest point. This concept is fundamental in various fields, such as economics for maximizing profits or minimizing costs.

Think of a hiker climbing a mountain. The highest peak they reach is analogous to a maximum point, while the lowest valley they pass through is similar to a minimum point.

🔹 First Derivative Test: Let 𝑓′(𝑥) = 0 at some point 𝑥 = 𝑐. • If 𝑓′(𝑥) changes from +ve to –ve at 𝑐, then 𝑓 has a local maximum at 𝑐. • If 𝑓′(𝑥) changes from –ve to +ve at 𝑐, then 𝑓 has a local minimum at 𝑐.

The First Derivative Test is a mathematical method used to identify local extrema of a function. The test involves evaluating the derivative of a function at a specific point. If the derivative changes from positive to negative, it indicates a peak, or local maximum, at that point. Conversely, if it changes from negative to positive, it indicates a trough, or local minimum. These changes illustrate how the slope of the function moves from increasing to decreasing, or vice versa.

Imagine driving up a hill. When your car is climbing, your speed (derivative) is positive. When you reach the top (the peak) and begin descending, your speed becomes negative. This change signifies that you have reached a maximum height. Similarly, if you drive into a valley, starting from a lower speed, your speed increases until you reach the lowest point before ascending again.

🔹 Second Derivative Test: If 𝑓′(𝑐) = 0 and: • 𝑓″(𝑐) > 0: Local Minimum • 𝑓″(𝑐) < 0: Local Maximum • 𝑓″(𝑐) = 0: Test fails, use First Derivative Test.

The Second Derivative Test is another method to determine whether a critical point is a maximum or minimum. After finding that the first derivative equals zero (indicating a potential extremum), we evaluate the second derivative at that point. If the second derivative is positive, it means the function is concave up at that point and thus indicates a local minimum. If the second derivative is negative, the function is concave down, signaling a local maximum. If the second derivative is also zero, the test is inconclusive, and the First Derivative Test should be applied.

Consider a roller coaster. The peaks represent local maxima (when the roller coaster is at the highest point) and the valleys represent local minima (when it's at the lowest point). If the track dips (positive second derivative), it suggests the coaster is entering a valley. If it rises (negative second derivative), it's going over a peak. If it's flat (zero), we need to check again to determine the type of point.

✅ Example: Find local maxima and minima of 𝑓(𝑥) = 𝑥³ − 6𝑥² + 9𝑥 + 2 Solution:

1. 𝑓′(𝑥) = 3𝑥² − 12𝑥 + 9
2. Set 𝑓′(𝑥) = 0: 3𝑥² − 12𝑥 + 9 = 0 ⇒ 𝑥 = 1,3
3. 𝑓″(𝑥) = 6𝑥 − 12 o 𝑓″(1) = −6 ⇒ Maximum at 𝑥 = 1 o 𝑓″(3) = 6 ⇒ Minimum at 𝑥 = 3

This example demonstrates how to identify local maxima and minima using both the first and second derivative tests. First, we find the first derivative of the function and set it to zero to find critical points. This leads us to potential points of local maxima or minima. After this, we compute the second derivative to evaluate these critical points. The signs of the second derivative at these points help us classify them as maxima or minima.

Imagine a business trying to find the optimal price for their product. The price point where the demand is highest may be considered a maximum, while points just before the price results in lower demand are analyzed as minima. Businesses often use mathematical models like this to fine-tune their pricing strategies for the best profit margins.

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