IB Mathematics Applications & Interpretation 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 $f^{\prime \prime }(x)$ 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 $f^{\prime \prime }(x)$ 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 $f^{\prime \prime }(x)=0$ for $f(x)=\ln (e^x-x)$.

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

Construct a sign chart for $f^{\prime \prime }(x)$ 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 $e^x(2-x)-1=0$.