Exactl\(y'\)

Calculus Level 4

\[f'(x)+\dfrac{f(x)}{x}-\sqrt{f(x)}=0\]

If a function \(f(x)\) satisfies the above equation and \(f(1)=12\), then which is of the following is true?

\(A. f(x)=\dfrac{x^3+12x^{\frac{3}{2}}\sqrt{3}-2x^{\frac{3}{2}}-12\sqrt{3}+109}{9x}\)

\(B. f(x)=\dfrac{x^3+12x^{\frac{3}{2}}\sqrt{3}-2x^{\frac{3}{16}}-12\sqrt{3}+109}{9x}\)

\(C.f(x)=\dfrac{x^3+12x^{\frac{5}{2}}\sqrt{3}-2x^{\frac{3}{2}}-12\sqrt{31}+109}{9x}\)

Clarification: \(f'(x)\) is the first derivative of the function \(f(x)\).

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