Final answer:
To find g(8) using the given information, we first need to find the general solution for f(x) by solving the differential equation. Then, we substitute f(x) and f'(x) into the expression for g(x) and use the given value of g(3) to find g(8).
Step-by-step explanation:
To find g(8), we need to use the given information for g(3). We know that g(x) = [f(x)]^2 + [f'(x)]^2. First, let's focus on finding f(x). The given equation f''(x) + f(x) = 0 is a second-order linear homogeneous differential equation with characteristic equation r^2 + 1 = 0. This equation has complex roots r = ±i. So, the general solution for f(x) is f(x) = c1*cos(x) + c2*sin(x), where c1 and c2 are constants.
Now, let's find f'(x) by differentiating f(x) with respect to x. f'(x) = -c1*sin(x) + c2*cos(x).
Now we can substitute f(x) and f'(x) into the expression for g(x): g(x) = [f(x)]^2 + [f'(x)]^2 = [c1*cos(x) + c2*sin(x)]^2 + [-c1*sin(x) + c2*cos(x)]^2 = c1^2 + c2^2.
Since we are given g(3) = 8, we have c1^2 + c2^2 = 8. We can't determine the exact values of c1 and c2, but we can use this equation to find g(8) by substituting c1^2 + c2^2 = 8 into g(x): g(8) = c1^2 + c2^2 = 8.