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

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

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

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

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

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

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

Construct a sign chart for $f''(x)=\frac{e^x(2-x)-1}{(e^x-x)^2}$ and use it to confirm concavity intervals.

Show by algebraic manipulation that $$ f''(x)=\frac{e^x(x-2)+x}{(e^x - x)^2}

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

Find the coordinates of the inflection point(s) 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 $e^x(2-x)-1=0$.

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