How do you sketch the graph by determining all relative max and min, inflection points, finding intervals of increasing, decreasing and any asymptotes given #f(x)=(x-4)^2/(x^2-4)#?

Question

Elijah Abram · Accepted Answer

Relative maximum #f(1)=-3# and relative minimum f(4)=0. Asymptotes : #uarr x=+-2 darr. POI : (5.034, 0.0500). f darr# : x in (1, 2) and (2, 4)# and #f uarr :, x in (-oo, -2), (-2, 1) and (4, oo) #

I took 30' for this answer. I should have a break. Please make another answer, without attempting to edit my answer. After some time, I would edit my answer, myself.. I have to repeat this because some rush to supply typo omissions of characters like #. I would do it myself. Logical and computational errors can be pointed out in the comment column and not by editing my answer. . graph{(x-4)^2/(x^2-4) [-30, 30, -15, 15]} x-intercept ( y = 0) : 4 y-intercept ( x = 0 ) : #-4# By actual division and resolving into partial fractions, #y=f = 1+1/(x-2)-9/(x+2)# As # x to 2 and -2, f to +-oo# #f'=-1/(x-2)^2+9/(x+2)^2# #=8((x-1)(x-4))/(x^2-4)^2# #=0, when x = 1 and 4, #f darr# intervals: #<0, x in (1, 2) and (2, 4)# and #f uarr# intervals : #>0, x in (-oo, -2), (-2, 1) and (4, oo) # Local minimum# #f(4)=0# Local maximum : #f(1)=-3# #f''=2/(x-2)^3-18/(x+2)^3=0#, when #9^(1/3)(x-2)=(x+2)#, giving #x=2(9^(1/3)+1)/(9^(1/3)-1)=5.7034, nearly# #f'''= -6/(x-2)^4+54/(x+2)^4 ne 0, at x = 5.034#, and here y = 0.05, nearly So, the POI is at x = 5.034, nearly

Ezra Adams · Answer

undefined To sketch the graph of $ f(x) = \frac{{(x-4)^2}}{{x^2-4}} $:

1. Find the critical points by setting the derivative equal to zero and solving for $ x $.
2. Determine the intervals of increasing and decreasing by analyzing the sign of the derivative in each interval.
3. Find the points of inflection by determining where the second derivative changes sign.
4. Identify any vertical asymptotes by finding where the denominator equals zero and the numerator doesn't.
5. Determine any horizontal asymptotes by analyzing the behavior of the function as $ x $ approaches positive and negative infinity.
6. Sketch the graph using the information obtained from steps 1-5.