Question

Encontre a expressão que define a função f, cujo grafico contem o ponto (4;5/3) e cuja derivada é f'(x)=...

Answer 1

$\displaystyle \sf f'(x) = \sqrt{x}(2\sqrt{x}-1) \\\\ f'(x) =2x-\sqrt{x} \\\\ \underline{integrando \ dos \ dois\ lados }: \\\\ f(x) = \frac{2x^2}{2}-\frac{\displaystyle x^{\frac{1}{2}+1}}{\displaystyle \frac{1}{2}+1}+C\\\\\\ f(x) = x^2 -\frac{x^{\frac{3}{2}}}{\displaystyle \frac{3}{2}} +C\\\\\\\ f(x) = x^2-\frac{2\sqrt{x^3}}{3}+C$

$\displaystyle \sf \underline{substituindo \ o \ ponto \ \left(4 , \frac{5}{3}\right) }: \\\\\\ f(4) = \frac{5}{3} \\\\\ \frac{5}{3} = 4^2-\frac{2\sqrt{4^3 }}{3} + C \\\\\\ \frac{5}{3}+\frac{2\sqrt{64}}{3}-16 = C \\\\\\ C = \frac{5+2.8-48}{3} \\\\\\ C = \frac{5+16-48}{3} \\\\\\ C = -9$

Portanto :

$\huge\boxed{\displaystyle \sf {\text f(x) =\text x^2-\frac{2\sqrt{\text x^3}}{3}-9 }\ }\checkmark$