​$$\text{Consider continuous solutions } f \text{ of the following integral equation in } [0,1]: \\[10pt]f^2(t) = 1 + 2 \int_0^t f(s) \, ds, \quad \forall t \in [0,1]. \\[10pt]\text{Which of the following statements is true?}$$​​

Correct option is C

​$$f^2(t) = 1 + 2 \int_0^t f(s) \, ds. \\[10pt]\text{Differentiating both sides:} \\[10pt]2f(t)f'(t) = 0 + 2f(t). \\[10pt]\text{Simplify:} \quad f'(t) = 1. \\[10pt]\text{Integrating:} \quad f(t) = t + C. \\[10pt]\text{Now, verify } f(t) \text{ by substituting into the equation:} \\[10pt](t + C)^2 = 1 + 2 \int_0^t (s + C) \, ds. \\[10pt](t + C)^2 = 1 + 2 \left( \frac{t^2}{2} + Ct \right). \\[10pt](t + C)^2 = 1 + t^2 + 2Ct. \\[10pt]\text{Expand } (t + C)^2: \quad t^2 + C^2 + 2Ct = 1 + t^2 + 2Ct. \\[10pt]C^2 = 1 \quad \Rightarrow \quad C = \pm 1. \\[10pt]\text{Thus, the two solutions are:} \\[10pt]f(t) = t + 1 \quad \text{or} \quad f(t) = t - 1. \\[10pt]\text{Conclusion: There are exactly two solutions.}$$​​