Question

Suppose that f is a continuous function that satisfies

$f(x)=x\int_{0}^{x} f(t)dt + x³$

For all x and that f(a) = 1, a ∈ R. Express f ' (a) in terms of a only:

Answer 1

$\text{Given $f(x) = x\int_{0}^{x}f(t)dt + x^3$ note that $\int_{0}^{x}f(t)dt = \frac{f(x)}{x} - x^2$, also $f(0) = 0$ then we will have the following: } \\ f'(x) = \int_{0}^{x}f(t)dt + x(f(x) - f(0)) + 3x^2 = \frac{f(x)}{x} - x^2 + xf(x) - xf(0) + 3x^2 \\ \qquad \,\,\, = \frac{f(x)}{x} + 2x^2 + xf(x) - x f(0) = \frac{f(x)}{x} + 2x^2 + xf(x) - x(0) = \frac{f(x)}{x} + 2x^2 + xf(x) \\ \implies f'(a) = \frac{f(a)}{a} + 2a^2 + af(a) = \frac1a + 2a^2 + a(1) = 2a^2 + a + \frac1a$