Question

Find the exact location of all the absolute extrema of the function f(3) = 24 - 12x3 Relative minimum at (9,-2187) Relative minimum at (9,-2187) and (0,0) Absolute minimum at (9,-2187) Absolute minimum at (0,9)

Answer 1

- 4-36 Now, check the sign of f(x) = 48 - 362² at each monca intonalin so; check the sign of 4% - 36x² at -osca.0:- Evaluate the derivate at a point on the interval. Take the point xat and plug it into 42 - 36x² so; 4 (7) ²–36 (+)2 = 4(-1) - 36(1) → 40 ( Negatue) Now; check the sign at ocagi- plug in x=5 +4G) ²-36 (52) 500 - goo -400 (Negative) Now; check the sign of 42² - 26% at So; plug x=10 - 4.10 -36.10² & 4 (1000) -36(10) A 4020-3600 $400 ( positive) Summary of monatone interwaks behaviour :- 4(128)-36(25) g<x<os - 92x20 OUCg 229 -o culo x=0 + sign Behaviour Decreasing saddle Decreasing minimum Increasing so; plug the entreme point a=0 into at 12x so; y= "1210? > y=0 Thus; saddle (0,0) Now; plug the extreme point x = 9 Ya 9 -12/93) - 6561-12(729) AY=6561-8748 & y7-2187 so; minimum: - ( 9, -287) Coption ci8 Correct)