1. **Problem:** Sketch and analyze the function $f$ with the given concavity and monotonicity properties. 2. **Key concepts:** - A function is **increasing** if its first derivative $f'(x) > 0$. - A function is **decreasing** if $f'(x) < 0$. - A function is **concave up** if its second derivative $f''(x) > 0$. - A function is **concave down** if $f''(x) < 0$. 3. **Part (a):** $f$ is concave up and increasing on $(-\infty, +\infty)$. - Since $f'(x) > 0$ and $f''(x) > 0$ for all $x$, the graph is increasing and bending upwards. - Example function: $f(x) = e^x$. 4. **Part (b):** $f$ is concave down and increasing on $(-\infty, +\infty)$. - Here, $f'(x) > 0$ but $f''(x) < 0$ for all $x$. - The graph increases but bends downward. - Example function: $f(x) = \ln(x)$ for $x > 0$ (restricted domain). 5. **Part (c):** $f$ is concave up and decreasing on $(-\infty, +\infty)$. - $f'(x) < 0$ and $f''(x) > 0$. - The graph decreases but bends upward. - Example function: $f(x) = -e^x$. 6. **Part (d):** $f$ is concave down and decreasing on $(-\infty, +\infty)$. - $f'(x) < 0$ and $f''(x) < 0$. - The graph decreases and bends downward. - Example function: $f(x) = -\ln(x)$ for $x > 0$ (restricted domain). **Summary:** - The sign of $f'(x)$ tells us if the function is increasing or decreasing. - The sign of $f''(x)$ tells us if the function is concave up or down. **Final answer:** - (a) $f'(x) > 0$, $f''(x) > 0$ - (b) $f'(x) > 0$, $f''(x) < 0$ - (c) $f'(x) < 0$, $f''(x) > 0$ - (d) $f'(x) < 0$, $f''(x) < 0$