Question

Find f.
f'(x)=(sqrt(x))(6+10x)
f(1)=11

Answer 1

\[f'(x)=\sqrt{x}(6+10x), f(1)=11\]

Answer 2

It is a matter of integrating \(f'\):\[f(x)=\int \sqrt{x}(6+10x)dx,\]\[f(x)=\int 6x^{1/2}dx+\int10x^{3/2}dx,\]\[f(x)=4x^{3/2}(x+1)+C.\]I combined the constants. Then use the initial condition to find \(C\):\[11=8+C\implies C=3.\]Therefore,\[f(x)=4x^{3/2}(x+1)+3.\]

Answer 3

thank you!