Função logarítmica

por ArielCouto Sáb 04 Mar 2017, 16:08

Alguém consegue resolver? Por favor.
Seja a função f dada por:

Função logarítmica Equayy10

Determine todos os valores de x que tornam f não negativa.

por superaks Sáb 04 Mar 2017, 16:42

Olá Ariel.

$\!\!\!\!\!\!\mathsf{(\ell og_35)\cdot\(\ell og_58^{x-1})+\ell og_34^{1+2x-x^2}-\ell og_32^{x(3x+1)}>0}\\\underline{\qquad\qquad\qquad\qquad}\\\\\mathsf{\ell og_58^{x-1}=\dfrac{\ell og_38^{x-1}}{\ell og_35}}\\\\\underline{\qquad\qquad\qquad\qquad}\\\\\mathsf{(\diagup\!\!\!\!\!\!\ell og_35)\cdot\Big(\dfrac{\ell og_3(2^3)^{x-1}}{\diagup\!\!\!\!\!\ell og_35}\Big)+\ell og_3(2^2)^{1+2x-x^2}-\ell og_32^{x(3x+1)}>0}\\\\\mathsf{\ell og_32^{3x-3}+\ell og_32^{2+4x-2x^2}-\ell og_32^{x(3x+1)}>0}\\\\\mathsf{\ell og_3\Big(\dfrac{2^{3x-3}\cdot2^{2+4x-2x^2}}{2^{x(3x+1)}}\Big)>0}\\\\\mathsf{\ell og_3(2^{3x-3+2+4x-2x^2-(3x^2+x)})>0}\\\\\mathsf{\ell og_3(2^{-5x^2+6x-1})}\\\\\mathsf{(-5x^2+6x-1)\cdot\ell og_3(2)>0}$

$\!\!\!\!\!\!\!\!\!\mathsf{\Delta=6^2-4\cdot(-5)\cdot(-1))}\\\mathsf{\Delta=36-20}\\\mathsf{\Delta=16}\\\\\mathsf{x=\dfrac{-6\pm\sqrt{16}}{2\cdot(-5)}}\\\\\\\mathsf{x^+=\dfrac{-6+4}{-10}\qquad\qquad\qquad x^-=\dfrac{-6-4}{-10}}\\\\\mathsf{x^+=\dfrac{-2}{-10}\quad\qquad\qquad\qquad x^-=\dfrac{-10}{-10}}\\\\\boxed{\mathsf{x^+=\dfrac{1}{5}}}\qquad\qquad\qquad\quad\boxed{\mathsf{x^-=1}}$

Como é uma concavidade voltada para baixo, os valores de x serão no intervalo:

[0,2; 1]