I really need HELP with this question, please help?

Question

Let f(x) = (x + 1)^2/(x – 1)^3. The f “(x) = 2x^2 + 20x +26/(x – 1)^5. Over what intervals is the curve concave up?

a) x < -5 - 2√3 and -5 + 2√3 < x < 1
b) -5 -2√3 < x < -5 + 2√3 and x > 1
c) x > 1
d) -5 - 2√3 < x < -5 + 2√3

Answer 1

You get the critical points where the curve changes concavity by setting f''(x) to 0:

[2x² + 20x + 26] / (x-1)^5 = 0
x² + 10x + 13 = 0
x² + 10x + 25 = 12
(x + 5) = ±√12
x = -5 ± 2√3

the value of f''(x) will change signs at the critical points, and f(x) is concave up when f''(x) > 0, so picking x = -2 < -5 + 2√3,
f''(-2) = [8 - 40 + 26]/ -3^5 = -6/-243 > 0, so concave up for x between -5 - 2√3 and -5 + 2√3. Also, when x > 1 (x = 1, the vertical asymptote, is another critical point), f''(x) > 0, so concave up.

That makes your answer b).