Question:

Step 1: Analyze the given integral and function properties. The function $f(x)$ is continuous for all real values of $x$, and it satisfies the property $\int_n^{n+1} f(x) dx = 2$ for all integers $n$. The integral to be evaluated is $-\int_3^5 f(|x|) dx$. Step 2: Simplify the integrand using the absolute value property. For the interval of integration $[3, 5]$, all values of $x$ are positive. Therefore, $|x| = x$ for $x \in [3, 5]$. The integral becomes: $$-\int_3^5 f(|x|) dx = -\int_3^5 f(x) dx$$ Step 3: Decompose the integral into unit intervals. The interval $[3, 5]$ can be decomposed into two unit intervals starting at integers: $[3, 4]$ and $[4, 5]$. $$-\int_3^5 f(x) dx = -\left[\int_3^4 f(x) dx + \int_4^5 f(x) dx\right]$$ Step 4: Apply the given property to each unit interval. Using the property $\int_n^{n+1} f(x) dx = 2$: For $n=3$: $\int_3^4 f(x) dx = 2$. For $n=4$: $\int_4^5 f(x) dx = 2$. Step 5: Calculate the final result. Substitute the values back into the decomposed integral: $$-\left[\int_3^4 f(x) dx + \int_4^5 f(x) dx\right] = -[2 + 2] = -4$$ The value of the integral is $-4$. Note: The provided "Correct Answer: b" (which is $\frac{35}{2}$) cannot be derived from the given problem statement and conditions. The calculation above is a direct and correct evaluation of the integral as stated in the question. If the intended answer is $\frac{35}{2}$, the problem statement or the given conditions must be different. Based on the literal interpretation of the question and the provided information, the result is $-4$.