3.2. Test Using Derivatives

Let 𝑓(𝑥) be a function defined on an interval 𝐼. • 𝑓(𝑥) is increasing on 𝐼 if for any two numbers 𝑥₁ < 𝑥₂ in 𝐼, we have 𝑓(𝑥₁) < 𝑓(𝑥₂). • 𝑓(𝑥) is decreasing on 𝐼 if 𝑓(𝑥₁) > 𝑓(𝑥₂).

To understand increasing and decreasing functions, we look at a function 𝑓(𝑥) defined on an interval 𝐼. The function is considered increasing on this interval if, for every pair of points 𝑥₁ and 𝑥₂ where 𝑥₁ is less than 𝑥₂, the value of the function at 𝑥₁ is also less than the value at 𝑥₂. In simpler terms, as you move along the x-axis from left to right, the function's output (y-values) keeps growing larger. Conversely, a function is decreasing if, as you move from left to right, the function's output is getting smaller.

Think of a hill: when you're climbing up, the height is increasing - this represents an increasing function. If you were to slide down the hill, your height would be decreasing, representing a decreasing function.

• If 𝑓′(𝑥) > 0 for all 𝑥 ∈ 𝐼, then 𝑓 is increasing on 𝐼. • If 𝑓′(𝑥) < 0, then 𝑓 is decreasing.

The derivative of a function, denoted as 𝑓′(𝑥), represents the rate of change of the function at any given point. If the derivative is positive (greater than zero), it indicates that the function's output is increasing at that point, meaning the graph of the function is slanting upwards as we move along the x-axis. Conversely, if the derivative is negative (less than zero), it means the function's output is decreasing, and the graph is slanting downwards. This provides a clear method to test whether a function is increasing or decreasing based on its derivative.

Imagine you're tracking a car's speed on a road. If the speed is positive (the car is accelerating), the car is moving faster and faster (an increasing function). If the speed is negative (the car is decelerating), the car is slowing down (a decreasing function).

✅ Example: Determine the intervals where 𝑓(𝑥) = 3𝑥² − 12𝑥 + 5 is increasing or decreasing. Solution: 𝑓′(𝑥) = 6𝑥 − 12 Set 𝑓′(𝑥) = 0 ⇒ 𝑥 = 2 • For 𝑥 < 2, 𝑓′(𝑥) < 0 → Decreasing • For 𝑥 > 2, 𝑓′(𝑥) > 0 → Increasing So, 𝑓(𝑥) is decreasing on (−∞, 2), increasing on (2, ∞)

In this example, we start with the function 𝑓(𝑥) = 3𝑥² − 12𝑥 + 5. First, we find the derivative, which is 𝑓′(𝑥) = 6𝑥 − 12. To find critical points, we set the derivative equal to zero: 6𝑥 − 12 = 0, which simplifies to 𝑥 = 2. Next, we evaluate the sign of the derivative to the left and right of 𝑥 = 2. For values less than 2 (like 𝑥 = 1), the derivative is negative, indicating that the function is decreasing. For values greater than 2 (like 𝑥 = 3), the derivative is positive, showing the function is increasing. Therefore, the function decreases on the interval (−∞, 2) and increases on (2, ∞).

Imagine a seesaw. When one side is lower than the pivot point (like x < 2), the seesaw tilts downwards (decreasing function). When the other side goes up past the pivot point (like x > 2), it starts lifting up and going higher (increasing function).

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