Question

Se F é uma é uma funçao de R em R tal que f(x)=3x³+x², determine a valor de : F^-1(5) + F(1)+F(-1)

Answer 1

$f(x)=3x^3+x^2$

Derivando:

$\dfrac{d}{dx}(3x^3+x^2)=x(9x+2)$

$f(x)=9x^2+2x$

$f(1)=9\cdot1^2+2\cdot1=9+2=11$

$f(-1)=9\cdot(-1)^2+2\cdot(-1)=9-2=7$

Seja $f^{-1}(x)=y$.

$x=9y^2+2y$

$y(9y+2)=x$

$y=\dfrac{x}{9y+2}$

$f^{-1}(x)=\dfrac{x}{9f^{-1}(x)}+2}$

$f^{-1}(5)=k$:

$k=\dfrac{5}{k+2}$

$k(k+2)5$

$k^2+2k-5=0$

$\Delta=2^2-4\cdot1\cdot(-5)=4+20=24$

$k=\dfrac{-2\pm\sqrt{24}}{2}=\dfrac{-2\pm2\sqrt{6}}{2}=-1\pm\sqrt{6}$

$f^{-1}(5)=-1+\sqrt{6}$

$f^{-1}(5)+f(1)+f(-1)=-1+\sqrt{6}+11+7=17+\sqrt{6}$


$f^{-1}(5)=-1-\sqrt{6}$

$f^{-1}(5)+f(1)+f(-1)=-1-\sqrt{6}+11+7=17-\sqrt{6}$