Respuesta :
[tex]\begin{gathered} f(x)=x^2+x-2 \\ f^{\prime}(x)=2x+1 \end{gathered}[/tex]
Let's use First-derivative test:
[tex]\begin{gathered} for\text{ x=0} \\ f^{\prime}(x)=0 \\ 2x+1=0 \\ x=-\frac{1}{2} \end{gathered}[/tex]So, -1/2 is a critical number of the function:
[tex]\begin{gathered} for\text{ } \\ x<-\frac{1}{2} \\ f(-2)=2(-2)+1=-4+1=-3 \\ -3<0 \end{gathered}[/tex]Therefore the function decreases over the interval:
[tex](-\infty,-\frac{1}{2})[/tex][tex]\begin{gathered} for \\ x>-\frac{1}{2} \\ f(1)=2(1)+1=3 \\ 3>0 \end{gathered}[/tex]Therefore the function increases over the interval:
[tex](-\frac{1}{2},\infty)[/tex]