Question

Decide the maclaurin series of the function, f(x), seen in the picture, of order 7.

Determine if f(x) has a local maximum or minimum in x = 0.

f(x) = = x² cos (sin 2x) + 2 – x2. Bestäm Maclaurinutvecklingen av f(x) med restterm 0(28). Avgör om f(x) har ett lokalt maximum eller minimum i x = 0. Beräkna f®)0).

Answer 1

+ 2 2 f'(o) = - f"(*) fra#Cool 810 2x ) -»*+2 from Carcos (81920) -- -* *2]'a 2x C- & Sin (sin 2x)) Cosze + Cos (Sin (21) ) ) fico) 42" sin(2x) sin(Sin (2x)) _407 Cos" (2x) Cos (sin (2x))- 8n sin (sin (2x)) Cosen +2 con sim tzw)) -? f(0) = $"(n) = 44** Sinza cos (2x) Cars (Smoczu) ) +8148.13 (son (und) Cos3 (21) + 8 x ~ Sims (Sin (2n)) Car (22) + 242 Sinj (2x) Sin(Sin (2x)) = 24 x Cos (22) Con (Sin (21)) - 12 Sin (Sin (2.)) Costa t" (0) = 0 f": -48* SDP (27) Cos CS-en)) -96 * Bin Cza) Sin (8in (2) Cos2 (21) . . 16x Sir (2) Sin(Sin (2x)) + 1622 Cost (2x) Cos (Sin (22)) a 64 av Cos? (2x) Cas (sinches) +192481n (22) Cos (2m2 cos (sin(20)) +642 Sin (Sin (2+)) los 3(2x) +64a Sin (Sirs (an) 2 Gs 2x +48 since Sin Esin (2x)) -48 Cast (21) Cos (sin (2x)) fico)=-48) frco=0, f"Colauad (0)=0 fres à pensa 2-224 + 2900 6! :: Payla (maclauni) Semes of arcos (Sm (2x)) -> 42 cepto n=2 is. ten) = 2- 234+ 30 3333 x vil 61 хь x 6

80 2=0 fio)= x Cos (Sinc2a)) = x+2 fluss -ex Cos 2x Bin CSinza) +2x COACsince)-22 -za Cosen.sin(Sim 22) —con (sintea)+} The extremum point obtained by equating f'(x)=0 and Ex cos z 8000 Sora)-cas(sipasu f"(a) the nature of function f"(x) = 4xYsincza 3–82 cos cza). Sin (Son (2x)) + (2-4x Cost (2.)) Cos (Sin (2x)) = 2 0 - 810) +(2-0) Co8Co)-2 f"(0) = 0 So the function has neither maxima not decides the at MO f.com mimima at x=0 be gafact frosse, which may maximum sh minimum