1. **Problem statement:** Given the graphs of a function $f$ and its derivative $f'$, answer the following: a) Find $f(2)$. b) Find $f'(2)$. c) Find the value of the derivative of $f$ at $x=0$. d) Find the slope of $G_f$ at point $A(-1|0.5)$. e) Find a value $a$ such that $f(a) = 1$. f) Find two values $b$ such that $f'(b) = 1$. 2. **Recall:** - $f(x)$ is the value of the function at $x$ (the $y$-coordinate on $G_f$). - $f'(x)$ is the slope of $f$ at $x$ (the $y$-coordinate on $G_{f'}$). 3. **Step-by-step solutions:** **a) Find $f(2)$:** From the graph $G_f$, at $x=2$, the function value is $0$. **Answer:** $f(2) = 0$. **b) Find $f'(2)$:** From the graph $G_{f'}$, at $x=2$, the value of $f'$ is approximately $0$ (the brown curve crosses near zero). **Answer:** $f'(2) = 0$. **c) Find $f'(0)$:** From $G_{f'}$, at $x=0$, the value of $f'$ is about $1$ (the brown curve is near $y=1$). **Answer:** $f'(0) = 1$. **d) Slope of $G_f$ at $A(-1|0.5)$:** The slope of $G_f$ at $x=-1$ is $f'(-1)$, which is the value of $G_{f'}$ at $x=-1$. From $G_{f'}$, at $x=-1$, $f'(-1) = 1$. **Answer:** slope at $A$ is $1$. **e) Find $a$ such that $f(a) = 1$:** From $G_f$, $f(x) = 1$ at approximately $x=0$ and $x=1$. **Answer:** $a = 0$ or $a = 1$. **f) Find two $b$ such that $f'(b) = 1$:** From $G_{f'}$, $f'(x) = 1$ at approximately $x=-1$ and $x=0$. **Answer:** $b = -1$ and $b = 0$. 4. **Summary:** $$ \begin{aligned} f(2) &= 0 \\ f'(2) &= 0 \\ f'(0) &= 1 \\ \text{slope at } (-1,0.5) &= 1 \\ f(a) = 1 &\Rightarrow a = 0, 1 \\ f'(b) = 1 &\Rightarrow b = -1, 0 \end{aligned} $$