Calculus Solution: Finding y(2) with Given Conditions

This problem involves finding the value of a function $y = f(x)$ at a specific point, given information about its derivatives and function values at other points. We use integration to find the function $f(x)$.

Step-by-Step Integration and Condition Application

1. Integrate the Second Derivative: We are given that $\frac{d^2y}{dx^2} = 0$. Integrating this once with respect to $x$ yields the first derivative: $ \frac{dy}{dx} = \int 0 \, dx = C_1 $ where $C_1$ is the constant of integration.
2. Apply the First Derivative Condition: The problem states that $\frac{dy}{dx} = 0$ when $x = 0$. Substituting these values into the equation from Step 1: $ 0 = C_1 $ Thus, the first derivative is $\frac{dy}{dx} = 0$.
3. Integrate the First Derivative: Integrating the first derivative $\frac{dy}{dx} = 0$ with respect to $x$ gives the function $y$: $ y = \int 0 \, dx = C_2 $ where $C_2$ is another constant of integration.
4. Apply the Function Value Condition: We are given that $y = 1$ when $x = 1$. Substituting these values into the equation from Step 3: $ 1 = C_2 $ So, the function is $y = 1$.
5. Determine y at x = 2: Since the function is determined to be $y = 1$ for all values of $x$, the value of $y$ at $x = 2$ is $1$.

Conclusion

Following the integration steps and applying the given conditions ($\frac{d^2y}{dx^2} = 0$, $\frac{dy}{dx} = 0$ at $x = 0$, and $y = 1$ at $x = 1$), we find that the function is $y=1$. Therefore, the value of $y$ at $x=2$ is 1.