Final answer:
To find f''(2), the second derivative of f at x = 2, the given known functional equation f(x) = (x²-1)f(2x) is differentiated twice, and the known values for f(4), f'(4), and f''(4) are utilized. The calculated value of f''(2) is -34.
Step-by-step explanation:
To calculate f''(2), we must differentiate the given functional equation f(x) = (x²-1)f(2x). In the provided case, we are given specific values for f(4), f'(4), and f''(4), which will help us determine the second derivative of f at x=2. First, let's take the derivative of both sides of the given functional equation: f'(x) = 2x*f(2x) + (x² - 1)*2*f'(2x). Now, let's find the second derivative by differentiating once more: f''(x) = 2*f(2x) + 2x*2*f'(2x) + 2*f'(2x) + 2x*(x² - 1)*2*f''(2x). Let's evaluate f''(x) at x=2 using the known values: f(4) = 2, f'(4) = 1, f''(4) = -2. This gives us: f''(2) = 2*f(4) + 2*2*2*f'(4) + 2*f'(4) + 2*2*(2² - 1)*2*f''(4), f''(2) = 2*2 + 2*2*2*1 + 2*1 + 2*2*(4 - 1)*2*(-2), f''(2) = 4 + 8 + 2 - 48. f''(2) = -34. Therefore, the second derivative of f at x=2 is -34.