$f(1)=?$

Calculus Level 4

Let the function $f(x)$ such that $f\left( 2 \right) =-\frac { 1 }{ 3 }$ and $f'(x) =x\left(f(x)\right)^{ 2 }$ with all real values of $x$ .

What is the value of $f(1)$ ?

-\frac { 2 }{ 3 }

-\frac { 2 }{ 9 }

-\frac { 11 }{ 6 }

-\frac { 7 }{ 6 }

1 solution

Guilherme Niedu
Jun 29, 2018

$\large \displaystyle \frac{dy}{dx} = xy^2$

$\large \displaystyle \frac{dy}{y^2} = x dx$

$\large \displaystyle \int_{- \frac13}^{y} \frac{dy}{y^2} = \int_2^x x dx$

$\large \displaystyle - \frac{1}{y} \Bigg |_{- \frac13}^{y} = \frac{x^2}{2} \Bigg |_2^x$

$\color{#20A900} \boxed{ \large \displaystyle y = - \frac{2}{2+x^2} }$

So:

$\color{#3D99F6} \boxed{ \large \displaystyle y(1) = -\frac23 }$