What are the number lines for f(x), f'(x) and f''(x) ?? showing the convacity for this derivative function belowthank you! ### Mathematical FunctionThe given mathematical function is expressed as:\[ f(x) = e^{-x} \cdot \frac{3}{x+1} \]#### ExplanationThis function, \( f(x) \), is a combination of an exponential function and a rational function. It includes the following components:- **Exponential Component**: \( e^{-x} \) - This represents an exponential decay function, where the base \( e \) is the mathematical constant approximately equal to 2.71828.- **Rational Component**: \( \frac{3}{x+1} \) - This describes a rational function with a numerator of 3 and a denominator of \( x+1 \).#### Properties- **Domain**: The function is undefined where the denominator equals zero. Thus, \( x \neq -1 \).- **Behavior**: The exponential part \( e^{-x} \) decreases as \( x \) increases, while the rational part \( \frac{3}{x+1} \) can vary significantly depending on the value of \( x \).This function's behavior is determined by the interplay between exponential decay and rational variation, and is commonly found in applications involving rates of change and asymptotic analysis.

Name: What are the number lines for f(x), f'(x) and f''(x) ?? showing the co
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: What are the number lines for f(x), f'(x) and f''(x) ?? showing the convacity for this derivative function belowthank you! ### Mathematical FunctionThe given mathematical function is expressed as:\[ f(x) = e^{-x} \cdot \frac{3}{x+1} \]#### Explanatio