1. **State the problem:** We are given the graph of $f'$, the derivative of $f$, and asked to determine all possible inferences about $f$ and $f'$ at $x=0$. 2. **Recall key concepts:** - The value of $f'(0)$ tells us the slope of $f$ at $x=0$. - If $f'(0) = 0$, $f$ may have a local max, min, or neither at $x=0$. - The sign of $f'(x)$ near $0$ indicates whether $f$ is increasing or decreasing. - A sign change of $f'$ from positive to negative at $x=0$ indicates a local maximum of $f$. 3. **Analyze the graph at $x=0$:** - From the graph, $f'(0)$ is approximately $-1$, so $f'(0) < 0$. - To the left of $0$, $f'(x) > 0$ (positive). - To the right of $0$, $f'(x) < 0$ (negative). 4. **Interpret these observations:** - Since $f'(x)$ changes from positive to negative at $x=0$, $f$ has a local maximum at $x=0$. - Because $f'(0) eq 0$ but negative, the slope of $f$ at $0$ is negative. - $f$ is increasing just before $0$ and decreasing just after $0$. 5. **Summary:** - $f'(0) \approx -1$ (negative slope). - $f$ has a local maximum at $x=0$ due to the sign change of $f'$ from positive to negative. - $f$ is increasing on an interval just left of $0$ and decreasing just right of $0$. **Final answer:** $$f'(0) \approx -1 < 0$$ $$\text{Sign change of } f' \text{ from } + \text{ to } - \text{ at } x=0 \implies f \text{ has a local maximum at } x=0$$ $$f \text{ is increasing for } x<0 \text{ near } 0 \text{ and decreasing for } x>0 \text{ near } 0$$