Key Concepts and Formulas

• Properties of Derivatives: If the second derivative of a function is zero, then the first derivative is a constant. If the first derivative is a constant, then the function itself is linear.
• Integration: The integral of a constant $k$ with respect to $x$ is $kx + C$, where $C$ is the constant of integration.
• Fundamental Theorem of Calculus: If $F'(x) = f(x)$, then $\int_a^b f(x) \, dx = F(b) - F(a)$.

Step-by-Step Solution

1. Analyze the given differential equation: We are given that $f''(x) - g''(x) = 0$. This can be rewritten as $f''(x) = g''(x)$.

   • Reasoning: This equation tells us that the second derivatives of the two functions are identical.

2. Integrate the differential equation once: Integrating both sides of $f''(x) = g''(x)$ with respect to $x$, we get: $\int f''(x) \, dx = \int g''(x) \, dx$ $f'(x) = g'(x) + k_1$ where $k_1$ is the constant of integration. This can be rearranged to: $f'(x) - g'(x) = k_1$

   • Reasoning: Since the second derivatives are equal, their antiderivatives (the first derivatives) must differ by a constant.

3. Determine the constant of integration $k_1$: We are given the values of the first derivatives at $x=1$: $f'(1) = 2$ and $g'(1) = 4$. Substituting these into the equation from Step 2: $f'(1) - g'(1) = k_1$ $2 - 4 = k_1$ $k_1 = -2$ So, the relationship between the first derivatives is: $f'(x) - g'(x) = -2$

   • Reasoning: We use the provided initial conditions for the derivatives to find the specific value of the constant.

4. Integrate the difference of the first derivatives: Now, we integrate the equation $f'(x) - g'(x) = -2$ with respect to $x$: $\int (f'(x) - g'(x)) \, dx = \int -2 \, dx$ $f(x) - g(x) = -2x + k_2$ where $k_2$ is another constant of integration.

   • Reasoning: Integrating the difference of the first derivatives gives us the difference of the original functions, which will be a linear function.

5. Determine the constant of integration $k_2$: We are given the values of the functions at $x=2$: $f(2) = 3$ and $g(2) = 9$. Substituting these into the equation from Step 4: $f(2) - g(2) = -2(2) + k_2$ $3 - 9 = -4 + k_2$ $-6 = -4 + k_2$ $k_2 = -6 + 4$ $k_2 = -2$ So, the expression for the difference of the functions is: $f(x) - g(x) = -2x - 2$

   • Reasoning: We use the provided function values at $x=2$ to solve for the constant of integration for the original functions.

6. Evaluate the difference of the functions at $x = \frac{3}{2}$: We need to find the value of $f(x) - g(x)$ when $x = \frac{3}{2}$. Using the expression derived in Step 5: $f\left(\frac{3}{2}\right) - g\left(\frac{3}{2}\right) = -2\left(\frac{3}{2}\right) - 2$ $= -3 - 2$ $= -5$

   • Reasoning: Substitute the specific value of $x$ into the derived expression for $f(x) - g(x)$ to find the required value.

Common Mistakes & Tips

• Confusing Constants of Integration: Remember that each integration step introduces a new constant of integration. Ensure you use different symbols (e.g., $k_1$, $k_2$) for these constants.
• Algebraic Errors: Double-check all arithmetic and algebraic manipulations, especially when dealing with negative signs and fractions.
• Understanding the Implications of $f''(x) = g''(x)$: Recognizing that this implies $f'(x) - g'(x)$ is a linear function is crucial. If $f''(x) = g''(x) = 0$, then $f(x)$ and $g(x)$ are both linear functions themselves.

Summary

The problem provides information about the second derivatives of two differentiable functions, $f(x)$ and $g(x)$, and specific values of their first derivatives and function values at certain points. By integrating the given condition $f''(x) - g''(x) = 0$ twice, we established a linear relationship for $f(x) - g(x)$. We used the provided derivative values to determine the constant of integration for the first derivatives, and subsequently used the function values to determine the constant of integration for the functions themselves. This allowed us to find the explicit expression for $f(x) - g(x)$ and evaluate it at $x = \frac{3}{2}$.

The final answer is \boxed{-5}.