Which derivative of f do we use to test for concavity? How do we find inflection points? The given mathematical function is:\[ f(x) = \ln (x^2 + 2x + 5) \]This function represents the natural logarithm of the quadratic expression \(x^2 + 2x + 5\). In calculus and algebra, natural logarithms (logarithms to the base \(e\), where \(e \approx 2.718\)) are often used to simplify complex equations and model exponential growth or decay.### Breaking Down the Equation:- **\( \ln \)**: The natural logarithm function.- **\( x^2 + 2x + 5 \)**: A quadratic expression.### Key Points for Understanding:1. **Natural Logarithm (ln)**: - The natural logarithm function \( \ln(x) \) is the inverse of the exponential function \(e^x\). - It is defined for all real numbers greater than zero. - The natural logarithm of a number is the exponent to which \(e\) must be raised to equal that number. 2. **Quadratic Expression**: - The expression inside the logarithm is a quadratic polynomial of the form \(ax^2 + bx + c\) where \(a = 1\), \(b = 2\), and \(c = 5\).### Understanding the Function's Behavior:- The function \(f(x)\) is defined for all real numbers \(x\) since \(x^2 + 2x + 5 > 0\) for all \(x \in \mathbb{R}\). This ensures that the argument of the natural logarithm is always positive, and hence \( \ln(x^2 + 2x + 5) \) is always defined.### Example Calculations:If we substitute specific values of \(x\) into \(f(x)\):- For \(x = 0\): \[ f(0) = \ln (0^2 + 2 \cdot 0 + 5) = \ln (5) \] - For \(x = 1\): \[ f(1) = \ln (1^2 + 2 \cdot 1 + 5) = \ln (8) \] This function provides a way to measure the logarithmic growth subject to the quadratic expression inside the logarithm.

Name: Which derivative of f do we use to test for concavity? How do we find
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: Which derivative of f do we use to test for concavity? How do we find inflection points? The given mathematical function is:\[ f(x) = \ln (x^2 + 2x + 5) \]This function represents the natural logarithm of the quadratic expression \(x^2 + 2x + 5\). In