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Chapter 3: Problem 43

For what intervals is \(f(x)=x e^{x}\) concave up?

Short Answer

Expert verified
The function is concave up on \((-2, \infty)\).

Step by step solution

01

Find the first derivative

To determine intervals of concavity, we first need the first derivative of the function. Given the function \( f(x) = x e^x \), we apply the product rule for differentiation: \( f'(x) = \frac{d}{dx}(x) \cdot e^x + x \cdot \frac{d}{dx}(e^x) = e^x + x e^x \). Therefore, the first derivative is \( f'(x) = (1 + x)e^x \).
02

Find the second derivative

Next, we need the second derivative, \( f''(x) \), to analyze concavity. Differentiate \( f'(x) = (1 + x)e^x \) using the product rule: \( f''(x) = \frac{d}{dx}(1 + x) \cdot e^x + (1 + x) \cdot \frac{d}{dx}(e^x) = e^x + (1 + x)e^x = (2 + x)e^x \). Thus, the second derivative is \( f''(x) = (2 + x)e^x \).
03

Determine where the second derivative is positive

The function \( f(x) \) is concave up where \( f''(x) > 0 \). Thus, we need to solve \((2 + x)e^x > 0\). Since \( e^x \) is positive for all real \( x \), we focus on \( 2 + x > 0 \). This inequality simplifies to \( x > -2 \).
04

Identify the interval of concavity

Based on Step 3, the interval on which the function \( f(x) \) is concave up is \( x > -2 \). In terms of interval notation, this means the concave up interval is \((-2, \, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is a fascinating branch of mathematics that deals with change and motion. It allows us to understand and predict how quantities vary. In the context of functions, calculus explores rates of change through derivatives and the accumulation of quantities with integrals. When we talk about a function like \( f(x) = x e^x \), calculus helps us dive deep into the behavior and trends this function displays along its curve. One essential aspect we investigate is how the shape of the curve bends or concaves at various points. This is where concepts like concavity come into play in calculus.
Derivatives
Derivatives are a fundamental tool in calculus, giving us a way to understand how a function changes at any given point. The process of finding a derivative is called differentiation. For the function \( f(x) = x e^x \), finding the first derivative, \( f'(x) \), involved using the product rule. This rule is essential when dealing with functions that are multiplied together. A derivative informs us about the slope or steepness of a function at any given point, and consequently, suggests how the function is increasing or decreasing.

  • First derivatives help determine intervals where functions are increasing or decreasing.
  • Second derivatives, which we also explore, inform us about the concavity of the function curve.
Understanding derivatives is crucial for a deeper grasp of how functions behave.
Concave Up
A function is said to be concave up on an interval if its graph lies above its tangents, resembling the shape of a cup facing upwards. Mathematically, this happens when the second derivative of the function, \( f''(x) \), is positive on that interval. For our example function \( f(x) = x e^x \), after calculating the second derivative as \( f''(x) = (2 + x)e^x \), we determine concavity by checking where this derivative is positive.

  • To find concave up regions, solve \( f''(x) > 0 \).
  • In our example, since \( e^x \) is always positive, we focus on \( 2 + x > 0 \).
This indicates where the function opens upwards visually, providing crucial insights into its behavior across different intervals.
Intervals of Concavity
Identifying intervals of concavity is all about determining where a function bends upwards or downwards on the graph. For the function \( f(x) = x e^x \), the second derivative test revealed that \( f(x) \) is concave up when \( x > -2 \). In interval notation, this is written as \((-2, \, \infty)\).

  • Intervals of concavity help in visualizing and predicting the behavior of functions.
  • This knowledge is also useful in optimization problems where maxima and minima need to be found swiftly.
Intervals where the second derivative is positive tell us exactly where the function takes on a specific concave shape, making this concept vital for graphing and interpreting functional data.

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Most popular questions from this chapter

Find the limit of the function as \(x \rightarrow \infty\). $$\frac{\sinh \left(x^{2}\right)}{\cosh \left(x^{2}\right)}$$

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