Let f be a twice differentiable function on R . If f'(0) = 5 , f''(0) = -8 and f(x) + ₀^x (x-t) f(t) dt = a e^x + b x , then the value of a^2 + b^2 is equal to _______.

Step-by-step solution

Given, f(x) + ₀^x (x-t) f(t) dt = a e^x + b x Substituting x = 0 , we get: f(0) = a + b Differentiating the given equation with respect to x using Leibniz's rule: f'(x) + ₀^x f(t) dt + x f(x) - x f(x) = a e^x - b x f'(x) + ₀^x f(t) dt = a e^x - b x Substituting x = 0 , we get: f'(0) = a Since f'(0) = 5 , we have a = 5 . Differentiating again with respect to x : f''(x) + f(x) = a e^x - b x Substituting x = 0 , we get: f''(0) + f(0) = a - b We are given f''(0) = -8 and we know f(0) = a + b . -8 + a + b = a - b 2b = 8