Concavity and Curve Sketching

Question

graph the function y = x^2 / x-5 by:

a. identifying the domain and any symmetries

b. finding the derivatives ... y' and y', x²-4

c. finding the critical points and identifying the function's behavior at each one

d. finding where the curve is increasing and where it is decreasing,

e. finding the points of inflection

f. determining the concavity of the curve

g. identifying any asymptotes

h. plotting any key points such as intercepts, critical points, and inflection points.

i. find the coordinates of absolute extreme points, if any

Answer 1

y = x²/(x - 5)

a. Domain: x =/= 5

b. y' = [2x(x - 5) - x^2]/(x - 5)² = (x² - 10x)/(x - 5)² = 1 - 25/(x - 5)²

y'' = 50/(x - 5)³

c. 1 - 25/(x - 5)² = 0

25/(x - 5)² = 1

25 = (x - 5)²

x = 0 x = 10 The function is neither decreasing or increasing at these points

d. The curve is increasing from negative infinity to zero & from ten to infinity not including ten and zero. The curve is decreasing from zero to five & from five to ten not including 5.

e. The second derivative never equals zero so technicallt there are no points of inflection. However, concavity changes before & after x = 5.

f. The curve is concave down from negative infinity to five & concave up from five to positive infinty. Neither include x = 5.

g. There are two asymptotes. One is at x =5 which is a vertical one. The other is at y = x + 5 which is a slant asymptote.

h. There is an x & y intercept at the origin.

i. There are no absolute extrema here since the cuve goes to both negative & positive infinity. There are relative maxima at x = 0 and minima at x =10.