Find and graph the coordinates of any local extreme points and inflection points

Question

Find and graph the coordinates of any local extreme points and inflection points of the function y = x^2 - 8 / x - 3, x = 3 Choose the correct answer regarding local extreme points. The function has a local minimum at (2,4) and a local maximum at (4,8). The function has a local maximum at (2,4) and a local minimum at (4,8). The function has no local extreme points. Choose the correct answer regarding inflection points. The function has inflection points at (2,4) and (4,8). The function has no inflection points. The function has an inflection point at (3,0). Choose the correct graph of y = x^2 - 8 / x - 3.

Explanation / Answer

f(x)=(x^2-8)/(x-3)

f'(x)=2x/(x-3) - (x^2-8)/(x-3)^2 = (2x^2-6x-x^2+8)/(x-3)^2=(x^2-6x+8)/(x-3)^2 = (x-4)(x-2)/(x-3)^2

f''(x)=(2x-6)/(x-3)^2 - 2(x^2-6x+8)/(x-3)^3=(2x^2+18-12x - 2(x^2-6x+8))/(x-3)^3 = (2)/(x-3)^3  

To find the local extrema f'(x)=0

(x-4)(x-2)/(x-3)^2=0

x=4,2

f''(2)= -2 <0 maxima at (2,4)

f''(4) =2>0 minima at (4,8)

Function has no inflection points.as f''(x)=0 gives no roots .

In graph D is the correct option checking the maxima minima and the behaviour of graph near x=3. At x=3+ f(x)goes to positive infinity and at x = 3- f(x) goes to negative infinity which is not the case in option B. so correct graph is D

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