Question

|(a) Consider the following function for > 0 f (x)= = -4x 48x (i) Find the stationary point(s) of this function. (3 marks) (ii) Is this function convex or concave? Explain why. (3 marks) (iii What type of stationary point(s) have you found? Include your reasoning. (4 marks) |(b) Show that ln(a) - a has a global maximum and find the value of a > 0 that maximises it. Do the same for ln(a) - a" where n is a positive integer. (8 marks)

Answer 1

-4x348 fe) Paints of the fumchom fta) poimbs those OTe чdx3+ 48 dx P -4*3x* t48} -12x2 48 ut f) -12 80 -12x-)v But 70 -2 ond 2. We com mot take *= -2 because fe) a defimed for x>o ?

(ii) Cemveritya fumetion (f"() i dekrmumеd Сиcmwig Ond (ts Second denvahve fumetion u Convex j fumchion u Concave e t amd tis Иeweva T, f"y)-0 в сad tnfuchion pernt Gткave to Gmvex. fume hon shonges foom baimg Where ncave Imfuehin Poin Convex Noww Let's f(n) -423+48 f(M) 12x2+ 48 "()= - 24х +0 Now, fx -24x0 fushiom peimt х -0 й the ence for Caiven) fumehion i Cneave

ufmad for avo Ima)-n f(a) Un(a)- a. -I0 New Let's find Semd derivahve oj femet'm fc). f"(a)= -0 . al as , f"() - - f(a) has the marimam valuu ata Simce fla) ha Tust m« ctical poins .Hance fla) has te otal mayimum at a f(a)= lna-oa" Now m ond ao f'a) ma mam 1-4D (-u)u - *7- (), f -n (m-) a a2 4-m-) 21 u an d aso valuus o a for all f'"(a) f (a) wil have maximum yalue at Hance,