Evaluate Differentiable Function Value Using Functional Equation and Integral Equation

To find the value of $g(2)$, we analyze the given functional equation and integral equation step-by-step.

Step 1: Determine the function $f(x)$

We are given the functional equation: $f(xy) = f(x) f(y) \quad \text{for all } x, y \in \mathbf{R}$

Substituting $y = 0$ into the equation, we get: $f(0) = f(x) f(0)$

Since $f(0) \neq 0$, we can divide both sides by $f(0)$: $f(x) = 1 \quad \text{for all } x \in \mathbf{R}$

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Step 2: Simplify the integral equation

Substituting $f(t) = 1$ into the given integral equation for $g(x)$: $x^2 g(x) = \int_{1}^{x} (t^2 - t g(t)) \, dt$

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Step 3: Differentiate to form a differential equation

Differentiating both sides with respect to $x$ using the Leibniz Rule: $\frac{d}{dx} \left[ x^2 g(x) \right] = \frac{d}{dx} \left[ \int_{1}^{x} (t^2 - t g(t)) \, dt \right]$

$2x g(x) + x^2 g'(x) = x^2 - x g(x)$

Rearranging the terms: $x^2 g'(x) + 3x g(x) = x^2$

Since $x \in [1, \infty)$, we have $x \neq 0$. Dividing both sides by $x$: $x g'(x) + 3g(x) = x$

Writing it in the standard linear first-order differential equation form: $g'(x) + \frac{3}{x} g(x) = 1$

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Step 4: Solve the differential equation

The integrating factor ($\text{I.F.}$) is: $\text{I.F.} = e^{\int \frac{3}{x} \, dx} = e^{3 \ln x} = x^3$

Multiplying the differential equation by $x^3$: $x^3 g'(x) + 3x^2 g(x) = x^3$ $\frac{d}{dx} \left[ x^3 g(x) \right] = x^3$

Integrating both sides with respect to $x$: $x^3 g(x) = \int x^3 \, dx$ $x^3 g(x) = \frac{x^4}{4} + C$

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Step 5: Find the constant of integration $C$

From the original integral equation, substitute $x = 1$: $1^2 \cdot g(1) = \int_{1}^{1} (t^2 - t g(t)) \, dt = 0 \implies g(1) = 0$

Now, substitute $x = 1$ and $g(1) = 0$ into the expression for $x^3 g(x)$: $1^3 \cdot 0 = \frac{1^4}{4} + C \implies C = -\frac{1}{4}$

Thus, the function $g(x)$ is given by: $x^3 g(x) = \frac{x^4 - 1}{4}$ $g(x) = \frac{x^4 - 1}{4x^3}$

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Step 6: Compute $g(2)$

Substitute $x = 2$ into the expression for $g(x)$: $g(2) = \frac{2^4 - 1}{4 \cdot 2^3} = \frac{16 - 1}{4 \cdot 8} = \frac{15}{32}$

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Conclusion

The value of $g(2)$ is $\frac{15}{32}$, which corresponds to option C.