Question

Let \( f : [0, \infty) \to [0, \infty) \) be continuous on \( [0, \infty) \) and differentiable on \( (0, \infty) \). If \[ f(x) = \int_0^x \sqrt{f(t)} \, dt, \text{ then } f(6) = \, \text{..........} \]

Hint:

When solving differential equations derived from integrals, first differentiate both sides, then solve the resulting equation.

Answer 1

Correct Answer: 9

Step 1: Differentiate both sides.
By differentiating the given equation \( f(x) = \int_0^x \sqrt{f(t)} \, dt \), we apply the Fundamental Theorem of Calculus to get: \[ f'(x) = \sqrt{f(x)} \]

Step 2: Solve the differential equation.
We solve \( f'(x) = \sqrt{f(x)} \). Squaring both sides, we get: \[ f'(x)^2 = f(x) \] This is a separable differential equation. Solving for \( f(x) \), we get: \[ f(x) = x^2 \]

Step 3: Conclusion.
Substituting \( x = 6 \) into \( f(x) = x^2 \), we get \( f(6) = 36 \). Thus, the correct answer is \( \boxed{36} \).