\begin{frame} \frametitle{2nd Midterm Exam - Review} \begin{exampleblock}{} Show that $f(x) = 2e^x + 3x + 15x^3$ has no tangent with slope $2$. \pause\bigskip We have: \begin{align*} f'(x) = 2e^x + 3 + 5x^2 \end{align*} \pause Note that \begin{align*} e^x \ge 0 &\quad\text{ for all $x$}\\ \mpause[1]{x^2 \ge 0} &\mpause[1]{\quad\text{ for all $x$}} \end{align*} \pause\pause and thus \begin{align*} f'(x) = 2e^x + 3 + 15x^2 \ge 3 \end{align*} \pause The slope of the curve $f(x)$ is $\ge 3$ everywhere.\\\pause Hence the curve cannot have a tangent with slope $2$. \end{exampleblock} \end{frame}