Question

Consider the following function. f(x) = 2x3 + 3.r? – 120. (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) (b) Find the open intervals on which the function is increasing or decreasing. (Select all that apply.) Increasing: (-9,-5) (-5, 4) (4,0) (-00,00) Decreasing: (-, -5) (-5, 4) (4,-) (-09, ) (C) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum (x, y) = ( relative minimum (x, y) = ( (d) Use a graphing utility to confirm your results.

Answer 1

we will use following condition to determine

* when f'(x) >0 => function increasing graph y=f(x) is moving upward

* when f'(x) <0 => function decreasing graph of y=f(x) moving downward

* at critical point f '(x) =0 or undefined , function changes from increasing to decreasing or vice versa at critical point   

* If function changes from increasing to decreasing at critical point the that critical point become point of maximum [ f"(x) <0 ]

* If function changes from decreasing to increasing at critical point the that critical point become point of minimum [f "(x) >0 ]

tirea bronchior to= 223 +3% -12on. differentiale from f. (m) = 2. dan (23) + 3 d (2) - 1104 t' (me 2x30 e34 in-12081 tl (m) = 60+60-120 Ra) At editical porno film ao 65+60-120=0 n-tn-2020 2+5n-un-2020 n (nts)- 4(ats) 20 (hts) (n-hao n+520 orn-hao n=-5 os n=4.. Coitical points or ho n=-5, n=4. bb). Ho increasity for fi (mzo 6nr60-12070 6 (0+5) (0-6) 7a (945) (0-6) 20 Sign chear -o sure ļ the 7 + 0 no (-os, -5) u (6,2 increcasino in (0,5) and (wo) option @ ga decrcasiz bo f1 (2) Lo (nts (0-4) Loon-G5,6) decreosis in (-5,4) option D

e At n=-5 to chanses from increasing to decreases so t (m attain maximaal nars . At nahtla chanses bromidecrecoin toinprecesio so to attain mininio at nah $(-5)=2(-5) 343 (-5) 120x5 = 425 +(S) = 264)3 + 3(5)~ _120xh = -304 relative masinoma (-5,425), minima a C4,-304)

-500。 6542 40 4,304 -5004