Find the open interval on which the function f(x)=x^2/x+1 is increasing or decreasing

Question

Find the open interval on which the function f(x)=x^2/x+1 is increasing or decreasing

Answer 1

I wanted to record a video for you but it is not working so I do my best to type it up.

In order to find the intervals on which the function is decreasing or increasing you must first find the critical points

1) f(x)=x²/(x+1)

1. the critical numbers/point are where the slope of the tangent line (first derivatite) is either 0 or undefined(horizontal slope or vertical asymptote)
2. Use the quotient rule to find the first derivative
3. summarized here : low d(high)- high d(low), all over low squared.
4. d ƒ(x)/dx=(x²+2x)/(x+1)²=[x(x+2)]/(x+1)²
5. This is the function to find the slope of the line, so set equal to 0.
6. To get dƒ/dx=0, only the numerator can be zero
7. x(x+2)=0
8. x=0 or x=-2
9. But there is a restricted value in the denominator, if x=-1 then the function is undefined
10. plug the x values back into the original function to find the y value of the points. from graph, you will see that x=-1 is a vertical asymptote.
11. see the graph at this link https://www.desmos.com/calculator/93lf5vxwqh
12. f(0)=0²/(0+1)=0 f(-2)=(-2)²/(-2+1)=-4
13. critical points are (0,0) and (-2,-4)
14. the function is increasing when d f(x)/dx>0 on the interval, decreasing when d f(x)/dx<0 on an interval.
15. from the graph linked above you can see when the function is decreasing and increasing by thinking about what the sign of the tangent line would be traveling from left to right. You can also pick points left and right of the critical points to test and plug the x values into the first derivative function to test or provide evidence.
16. From the graph
17. the slope of the tangent line is positive(increasing) from (-∞,-2), then negative (decreasing) from (-2,-1), and negative(decr.) again from (-1,0) and positive (incr.) from (0,∞)

Answer 2