Question

1. Consider the curve given by the function f(x) = -4.83 27(x + 1)2 You are given that -4x²(x +3) - 8.1 f'(x) = and f"(x) = 27(x + 1)3 9(x + 1)4 Compile the following information about f(x) and its graph. Show your work to justify your answers to parts (f), (g), (h), (i) and (j). Otherwise, give answers only. Answer "NONE” if the function does not display a feature listed. 1] (a) Domain of f (b) x and y coordinates of x-intercepts of f 1] (C) x and y coordinates of y-intercepts of f 1] (d) Equation(s) of vertical asymptote(s)

Answer 1

1. Givon that - 4x3 means =o in => 1S in thon f(x) = 27 (1+12 Ta] Domain of f is IR-{-1} [b] d-inturcepts of f y=0 - 4 x3 x=0 70%+1) then Coordinatus (0,0). [0] y- intercepts of f means -473 q=0 07 (4+1) coordinatus is cool [d] The line x=L is the vertical asymptote of function if the limit of the 27%+12 function at this point is infinite. So find the points where the denaminator equals o check them at 2:-1, Jim 4x3 (- ༡༡ (༡ཤད 27-1 Hunce [f] For cinitical points put HE - 4x2 (x+3) 27(x+1)3 473 1 and de-1 t's a vertical e symptes the 2-3 then a= and 3 are critical points.

> o Hence it is and for را -) [A] (9 71002 -422 (x+3) 27 (x+1) for -3<x< 1 increasing function in (-3,-1) f'alto and for C-00,-3 Honce for the intervas (-os, 23] and (-1,00) This is a decreasing decreasing function Ash The above discussion shows that f hang only lacad minimum point which because Is -3 eit X=-3 f"(x) >0 ,