1. **State the problem:** Given the function $f(x) = 2x - 1$, find the first and second derivatives, identify the correct graph for $f$ and $f''$, and determine intervals where $f$ is concave up or concave down. 2. **Find the first derivative $f'(x)$:** The derivative of $f(x) = 2x - 1$ with respect to $x$ is found using the power rule: $$f'(x) = \frac{d}{dx}(2x) - \frac{d}{dx}(1) = 2 - 0 = 2$$ 3. **Find the second derivative $f''(x)$:** The second derivative is the derivative of the first derivative: $$f''(x) = \frac{d}{dx}(2) = 0$$ 4. **Interpret the second derivative:** Since $f''(x) = 0$ for all $x$, the function has zero concavity everywhere, meaning it is a straight line with no curvature. 5. **Identify the graph:** The graph of $f$ is a blue line with positive slope (2), and $f''$ is a red horizontal line at $y=0$. This matches Graph A. 6. **Determine concavity intervals:** Since $f''(x) = 0$ everywhere, $f$ is neither concave up nor concave down on any interval. - $f$ is concave up on: DNE - $f$ is concave down on: DNE