Verified Answer :

LammettHash

Assuming you mean

\(f'(x)=\sqrt x(3-10x},\,f(1)=9\)

Integrate both sides to get

\(\displaystyle\int f'(x)\,\mathrm dx=f(x)=\int\sqrt x(3-10x)\,\mathrm dx\)

You have

\(f(x)=\displaystyle\int\sqrt x(3-10x)\,\mathrm dx=\int(3x^{1/2}-10x^{3/2})\,\mathrm dx=2x^{3/2}-4x^{5/2}+C\)

Since \(f(1)=9\), you have

\(9=2\times1^{3/2}-4\times1^{5/2}+C\implies C=11\)

so the particular solution is

\(f(x)=2x^{3/2}-4x^{5/2}+11=2x^{3/2}(1-4x)+11\)

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