Problem 16 - Calculus

Question

If \(d f / d x\) is negative, is \(f(x)\) always negative? Is \(f(x)\) negative for large \(x ?\) If you think otherwise, give examples.

Step-by-Step Solution

Answer

A negative derivative means \(f(x)\) is decreasing; it can be positive or negative. \(f(x)\) often becomes negative for large \(x\).

Step 1: Understanding the Meaning of a Negative Derivative

If \( \frac{df}{dx} \) is negative, it means that the function \( f(x) \) is decreasing. The slope of the tangent to the function at any given point is negative, indicating a downward trend. However, this does not directly imply that \( f(x) \) itself is negative.

Step 2: Example of Decreasing Functions

Consider a function that is decreasing but not necessarily negative, such as \( f(x) = -x + 10 \). The derivative \( \frac{df}{dx} = -1 \) is negative, showing that the function is decreasing. However, \( f(x) \) will be positive as long as \( x < 10 \). Thus, a negative derivative can result in a positive \( f(x) \).

Step 3: Assessing Large Values of \(x\)

To determine if \( f(x) \) is negative for large \( x \), we need to analyze the behavior of the function as \( x \) approaches infinity. Taking our example \( f(x) = -x + 10 \), as \( x \) gets larger, \( f(x) \) becomes more negative. Hence, \( f(x) \) is indeed negative for large \( x \).

Step 4: Conclusion

While a negative derivative implies a decreasing function, it does not mean that \( f(x) \) must be negative. However, for many functions like linear functions with a negative slope, \( f(x) \) can become negative when \( x \) is sufficiently large.

Key Concepts

Negative Derivative and its Implications

When the derivative of a function, denoted as \( \frac{df}{dx} \), is negative, it tells us something crucial about the function's behavior. Specifically, it indicates that the function \( f(x) \) is decreasing. But what does this mean?

• A negative derivative implies that the slope of the function is moving downwards. Imagine a hill that slopes downwards, this hill visually represents a function with a negative derivative.
• Even though the function is decreasing, it doesn't automatically mean that the actual values of the function \( f(x) \) are negative. These are two distinct concepts. A decreasing trend doesn't dictate the sign of the function's values.
• For example, consider the function \( f(x) = -x + 10 \). Here, \( \frac{df}{dx} = -1 \), indicating that the slope is negative, so the function is indeed decreasing. However, depending on the value of \( x \), the output \( f(x) \) can still be positive.

By understanding that a negative derivative simply reflects a downwards trend, we can prevent the common mistake of assuming negativity in the function's output.

Decreasing Function Explained

A function is termed decreasing when it consistently goes downwards as you move along the X-axis. This concept closely ties in with our previous discussion on negative derivatives.

• When \( \frac{df}{dx} \) is negative, it indicates a consistent downward movement of the function. Imagine skating down a slope, where each movement takes you a little lower.
• This doesn't mean the values of the function \( f(x) \) are themselves negative. Instead, it means the function consistently outputs smaller or lesser values."
• Taking our previous example, \( f(x) = -x + 10 \), shows a decreasing pattern as the situation consistently declines with increasing \( x \).

The key tack to understanding decreasing functions is focusing on the trend, not just the function's current position on the axis.

Understanding Function Behavior for Large \( x \)

Have you ever wondered what happens to a function when its variable, \( x \), gets very large? This is often crucial for understanding the function's broader behavior.

• As \( x \) grows towards infinity, it's important to observe whether \( f(x) \) trends towards positive or negative values, or stabilizes at some specific level.
• In many cases, like our example function \( f(x) = -x + 10 \), as \( x \) increases beyond certain boundaries (here it's 10), \( f(x) \) tends to become negative because \( -x \) dominates the positive constant. This makes the function trend towards negative values for large \( x \).

Understanding behavior at extremes helps in grasping the full picture of the function, allowing us to predict its long-term patterns effortlessly.