Problem 48 - Calculus

Question

Explain what is wrong with the statement. The function \(y=e^{-0.25 x}\) is decreasing and its graph is concave down.

Step-by-Step Solution

Answer

The function is decreasing, but the graph is concave up, not concave down.

Step 1: Identify Function Characteristics

The function provided is \(y = e^{-0.25x}\), which is an exponential function. It's important to recognize the basic characteristics of exponential functions to proceed with the analysis.

Step 2: Determine Increasing or Decreasing

To determine whether the function \(y = e^{-0.25x}\) is increasing or decreasing, we check the coefficient of \(x\) in the exponent of \(e\). Since \(-0.25\) is negative, as \(x\) increases, \(y\) decreases. Thus, the function is decreasing.

Step 3: Differentiate the Function

Find the first derivative to confirm the decreasing nature of the function and analyze its concavity: \[y' = \frac{d}{dx}(e^{-0.25x}) = -0.25e^{-0.25x}\]Since \(y' < 0\), it confirms that the function is decreasing.

Step 4: Analyze Concavity with Second Derivative

To determine concavity, find the second derivative:\[y'' = \frac{d^2}{dx^2}(e^{-0.25x}) = \frac{d}{dx}(-0.25e^{-0.25x}) = (0.25^2)e^{-0.25x} = 0.0625e^{-0.25x}\]Since \(y'' > 0\), the concavity is upwards, indicating the function is concave up, not concave down.

Step 5: Conclusion on Concavity

Because \(y''\) is positive, the function is actually concave up, not concave down as initially stated in the problem. Thus, there is an error in the statement regarding the direction of the concavity.

Key Concepts

Decreasing Functions

Exponential functions often exhibit behaviors like increasing or decreasing based on their exponential coefficients. For the function \( y = e^{-0.25x} \), we first need to look at the exponent's coefficient, which is \(-0.25\).
This negative coefficient implies that as \( x \) grows, the value of \( y \) shrinks. This means the function is decreasing. The rule of thumb is that if the exponent has a negative coefficient, the exponential function typically decreases.
Understanding these traits is crucial because they help identify whether a function grows or shrinks over a specific interval. In the context of this exercise, since the statement initially got the concavity wrong, focusing on the decrease correctly supports part of their analysis.

Concavity Analysis

Concavity in functions shows whether a graph curves upward or downward. It relies on the second derivative. For the function \( y = e^{-0.25x} \), determining its concavity tells us how the slope behaves over the function's domain.
By taking the second derivative \( y'' = 0.0625e^{-0.25x} \), which is positive, we learn that the graph is concave up. This means that the slope of the tangent line to the curve increases over time as the function progresses. If the second derivative is positive, the curve is bending upwards like a cup. Conversely, a negative second derivative signals concave down.

• Concave Up: \( y'' > 0 \)
• Concave Down: \( y'' < 0 \)

This distinction is vital as the exercise identifies an error: the function is not concave down, contradicting the original statement.

First Derivative Test

The first derivative test helps confirm whether a function is increasing or decreasing. By differentiating the function \( y = e^{-0.25x} \) to find \( y' = -0.25e^{-0.25x} \), we can analyze its behavior.
A negative first derivative confirms that the function is decreasing over its entire domain. This matches our expectations based on the negative exponent, reinforcing the analysis that the function becomes smaller as \( x \) increases.
In summary:

• If \( y' > 0 \): Function is increasing
• If \( y' < 0 \): Function is decreasing

The first derivative test provides an additional tool for understanding how the function's output changes as \( x \) changes. The zero or sign of \( y' \) gives clear evidence of whether the function rises or falls.

Second Derivative Test

Using the second derivative test allows us to examine not just the curvature of a graph, but also locate and identify possible inflection points. For our function \( y = e^{-0.25x} \), with its second derivative \( y'' = 0.0625e^{-0.25x} \) being positive, we notice it never changes sign.
This tells us that throughout its domain, the function remains concave up, avoiding any inflection points where the direction of concavity would switch. Using this second derivative, which is always greater than zero, we confidently state that the function cannot be concave down anywhere.
Key takeaways from the second derivative test include:

• If \( y'' > 0 \): Function is concave up
• If \( y'' < 0 \): Function is concave down
• Inflection Point: Exists where \( y'' = 0 \) and it changes sign

Thus, correcting the error in the original statement, it's clear the graph stays concave up rather than switching concavity.