First derivative test and curve sketch analysis

Question

y= (x-5)/(x^2)

sketch the curve using a first derivative test, second derivative test, and a curve sketch analysis.

I thought I had found that the critical values from finding the derivative and setting it equal to zero were 0 and 10 but then when i watched the homework help video and got really confused. if anyone could help clarify what the critical values to use in the 1st derivative test would be great! Thanks in advance!

Answer 1

1. One of the critical values is obviously x = 0, where y = -5/0 is a singularity with negative infinity.
2. y' = -(x-10)/x³, so when x = 10, y' = 0. However, the y' = 0 does not always mean a maximum or minimum, it might also be a inflection point. Thus we need to check the second derivative at those points.
3. y''= 2(x-15)/x⁴, when x = 10, y'' < 0, so y(x=10) is the maximum.
4. when x →+∞, y→ +0; when x → −∞, y→ −0;
5. so the curve has a singularity x = 0 and y→ −∞ from both side, on the left side of the singularity, y increase to 0 when x approaches −∞; on the right side, y increases and reach 0 at x = 5, and keep increasing and reach its maximum at x = 10 and then decreases to 0 when x approaches +∞.

Answer 2

desmos.com/calculator/gd6yfps89l

Did not get a chance to employ the 2nd derivative test or tests for concavity, but hopefully this gets you started in the right direction. Be sure to pause video and take notes.