Question

4. Find all the intervals over which f(x) = x²+x-2 a. (-infinity, infinity) b. (-infinity,0) (-infinity, 2), (1, infinity) d. (-infinity,-2), (-2,1),(1, infinity) e. none of these

Answer 1

`\begin{gathered} f(x)=x^2+x-2 \\ f^(\prime)(x)=2x+1 \end{gathered}`

Let's use First-derivative test:

`\begin{gathered} for\text{ x=0} \\ f^(\prime)(x)=0 \\ 2x+1=0 \\ x=-(1)/(2) \end{gathered}`

So, -1/2 is a critical number of the function:

`\begin{gathered} for\text{ } \\ x<-(1)/(2) \\ f(-2)=2(-2)+1=-4+1=-3 \\ -3<0 \end{gathered}`

Therefore the function decreases over the interval:

`(-\infty,-(1)/(2))`
`\begin{gathered} for \\ x>-(1)/(2) \\ f(1)=2(1)+1=3 \\ 3>0 \end{gathered}`

Therefore the function increases over the interval:

`(-(1)/(2),\infty)`