Question Type 3: Determining the regions of concavity and its type for a function Exercises

Determine the intervals on which the function $f(x) = \ln(e^x - x)$ is concave up and concave down.

Find the coordinates of the inflection points of $f(x)=\ln(e^x - x)$.

Find the second derivative $f''(x)$ of the function $f(x) = \ln(e^x - x)$.

Describe the behavior of the concavity of $f(x) = \ln(e^x - x)$ as $x \to \pm\infty$.

Evaluate $f''(x)$ at $x = -1$, $x = 0$, and $x = 2$, and state the concavity of $f(x) = \ln(e^x - x)$ at each of these points.

Solve the equation $f''(x)=0$ for $f(x)=\ln(e^x - x)$.

The second derivative of a function $f$ is given by $f''(x) = \frac{e^x(2-x)-1}{(e^x-x)^2}$, for $x \in \mathbb{R}$.

Construct a sign chart for $f''(x)$ and determine the intervals where the graph of $f$ is concave up and concave down.

Prove that $f(x)=\ln(e^x - x)$ has exactly two inflection points.

Find the first derivative of $f(x) = \ln(e^x - x)$.

Determine the domain of $f(x)=\ln(e^x - x)$.

Use two iterations of Newton–Raphson starting at $x_0=0$ to approximate the left-hand root of $e^x(2-x)-1=0$.