Question

#4

precision Problem 4. Discuss the function (domain, even, odd, limo+ f(), vertical asymptotes, intercepts, positivity, increasing, decreasing, maxima and minima, concave up, concave down, points of inflection, and finally a sketch of the graph). [30 pts, 15 min] (a) f(x) = (x + 1)(2x + 1); f'(x) = 2(1 + x)º(11x + 6); f"(x) = 10(x + 1) (22r +13) (b) f(x) = -100e-4; '(x) = (100 - x)299e-4; 1"(x) = (x - 110) (r - 90)2-98e-1

Answer 1

as f (x+1)! (2x+1) domain: IR flag is a polynomal) fl-x) = (-x+1) 10 (-2x+ 1) = -f(x) f(n) = Not even and not odd im fin) = lim fln) = Fool 24-0 attoo 4- intercepts! igazo Ho) = 1, il flm)=0) So, x-intereepts! (-1,02, (-1,0) and 4-intercept - Co, 12. vestical asymptots: None, as fin is defined Everywhere. d'(n) =0 = x= -0 1-1. on -ooLXla f'(x)70; (as t'(x) 70 & increasing) -1<x<- 6 it'(mco (as flnico a becere asing) - 6 L XL; fla27o. go by first derivative rest. local minima . ( -6 1-9765625) and maxima (-1,02 increasing on a (-00,-1) U (-610) decreasing on : (-1,-6) To find inflection points: fon2=0 =) X=-1, -13 s inflection points (-1 and -13/22) 22 as f is concare cup when f '(x) 70 fis concave down if f"(x) co.

-oocaan f(n)to -LAC-13; f(n) co 22 22 -13 caco , f(n) 7o concave upward on 1-13,00) a Concave down wold on

by time = 2 100 en as ex 8 x 100 defined everywhere e domain is lih as fl-x) = geloo ex. & f(x) on f(n) & Not even nom odd. lim x 10 ex = 10 j cum f(n) = 100 ato 27-00 as Hom) is defined every where so, Vectical asymptote is f None. Now, f(n)=0 = x=100, x=0

for-wocxco, f'(n) co for olxelo "flnlzo for low araw, film o and f is increasing decreasing on Now, points of infection! so, by first derivative test local minina (oro) local maxima 1100, 100100) eco on (0,100) and C-oo, olu Clow, ao 2. flm 1=0=1/x = 110,90 ono creo i "(alto inoLaLao i falroc on Go LX <110 i finto en 110LH Log f(n)70 con care upward on (ano,O) 000,90) (110 ) concave down ward on (90,110)