Key Concepts and Formulas

• Antiderivatives: Finding a function given its derivative involves integration (finding the antiderivative).
• Power Rule for Integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.
• Initial Value Problems: Using given conditions (e.g., a point on the graph, the slope of the tangent at a point) to determine the constants of integration.
• Derivative as Slope: The derivative $f'(x)$ represents the slope of the tangent line to the curve $y=f(x)$ at the point $x$.

Step-by-Step Solution

Step 1: Find the First Derivative, $f'(x)$

We are given the second derivative: $f''(x) = 6(x - 1)$ To find the first derivative $f'(x)$, we integrate $f''(x)$ with respect to $x$: $f'(x) = \int f''(x) \, dx = \int 6(x - 1) \, dx$ Expanding the expression inside the integral: $f'(x) = \int (6x - 6) \, dx$ Integrating term by term: $f'(x) = \int 6x \, dx - \int 6 \, dx$ Using the power rule for integration: $f'(x) = 6 \cdot \frac{x^2}{2} - 6x + C_1$ $f'(x) = 3x^2 - 6x + C_1$ Here, $C_1$ is the constant of integration.

Step 2: Use the Tangent Information to Determine $C_1$

We are given that the tangent to the graph at $(2, 1)$ is $y = 3x - 5$. The slope of this tangent line is $3$. Therefore, $f'(2) = 3$. Substituting $x = 2$ into the expression for $f'(x)$: $f'(2) = 3(2)^2 - 6(2) + C_1 = 3$ $12 - 12 + C_1 = 3$ $C_1 = 3$ Thus, the first derivative is: $f'(x) = 3x^2 - 6x + 3$ Factoring out a 3: $f'(x) = 3(x^2 - 2x + 1)$ $f'(x) = 3(x - 1)^2$

Step 3: Find the Function, $f(x)$

To find $f(x)$, we integrate $f'(x)$ with respect to $x$: $f(x) = \int f'(x) \, dx = \int 3(x - 1)^2 \, dx$ Let $u = x - 1$, so $du = dx$. Then, the integral becomes: $f(x) = \int 3u^2 \, du$ Using the power rule for integration: $f(x) = 3 \cdot \frac{u^3}{3} + C_2$ $f(x) = u^3 + C_2$ Substituting back $u = x - 1$: $f(x) = (x - 1)^3 + C_2$ Here, $C_2$ is another constant of integration.

Step 4: Use the Point Information to Determine $C_2$

We are given that the graph passes through the point $(2, 1)$. This means $f(2) = 1$. Substituting $x = 2$ into the expression for $f(x)$: $f(2) = (2 - 1)^3 + C_2 = 1$ $(1)^3 + C_2 = 1$ $1 + C_2 = 1$ $C_2 = 0$ Thus, the function is: $f(x) = (x - 1)^3$

Step 5: Match the Function to the Options

Comparing our result $f(x) = (x - 1)^3$ to the given options, we see it matches option (B).

Common Mistakes & Tips:

• Remember the Constants of Integration: Always add a constant of integration after each indefinite integral.
• Use Initial Conditions Carefully: Ensure you are using the correct initial conditions for $f(x)$ and $f'(x)$.

Summary:

To find the function $f(x)$ given its second derivative and initial conditions, we performed two successive integrations. We integrated $f''(x)$ to find $f'(x)$, using the tangent's slope at a given point to find the first constant of integration. Then, we integrated $f'(x)$ to find $f(x)$, using the point the graph passes through to find the second constant of integration. This led us to the function $f(x) = (x-1)^3$.

The final answer is \boxed{(x - 1)^3}, which corresponds to option (B).