Đáp án:

\(6 = \int_{0}^{2} \left(\frac{f'(x)}{\sqrt{f(x)}}\right)^2 dx.\)

Tìm \(k\) sao cho: \(\int_{0}^{2} \left[\frac{f'(x)}{\sqrt{f(x)}} - k\right]^2 dx = 0.\)

\(\Leftrightarrow \int_{0}^{2} \frac{[f'(x)]^2}{f(x)} dx - 2k \int_{0}^{2} \frac{f'(x)}{\sqrt{f(x)}} dx + k^2 \int_{0}^{2} dx = 0.\)

\(\Rightarrow 18 - 4k \sqrt{f(x)}\big|_{0}^{2} + k^2 x\big|_{0}^{2} = 0.\)

\(\Rightarrow 2k^2 - 12k + 18 = 0 \quad \Leftrightarrow k^2 -6k +9 =0  \quad \Leftrightarrow k = 3.\)

Suy ra: \(\frac{f'(x)}{\sqrt{f(x)}} = 3 \quad \Rightarrow \quad 2 \sqrt{f(x)} = 3x + C.\)

\(f(0) = 0 \Rightarrow C = 0.\)

\(\Rightarrow \sqrt{f(x)} = \frac{3x}{2} \quad \Rightarrow \quad \sqrt{f(1)}= \frac{3}{2} \Rightarrow f(1) = \frac{9}{4} \Rightarrow \boxed{A}\)