Question

Use the First Derivative Test to find the relative extrema of the function, if they exist. f(x) = x^4 - 2x^2 + 5

Answer 1

fun 1 Pg - 1 Solution first Derivative test for nelestive extremas suppore that cis critical point of con timour of f, cnd fir differen bubie at every point in some interval. contuining c except possibly at c itself. Moving acron from left to right, (1) IF f' chunger sign from ove to the ait c, then If hus relative minimum at. C. (2) If I changes tve to ve dit c, then of hus relative maximum at c. (3) If f does not change then that no local estremas arc. fal a 2" - 20? +5 foi = 4x² - 4x 11101_50. 4903-40 401622-1) = 0 40112 - 1)(a +12 20. 50, 020 7-1, T2-1

60, 110-27 EMO) -4 + 2 1) 12 ए (-0, -1) F'(-2) = 46-233-46-2) = -32 +9 = -24 <0 (-1,0) f(-1) 46-312) 3 - 414): :(0,1) 2 / 0 f'(712) = 4(!)3=461) - 4 - 30 (1,0) f'(2) 4(2)3 - 4(2) 2 d 8 32-8 24 0 O (1, + o 4 (00.,-1). $1,0)/2=0 10,1)|x=1 sign Behaviorl Decreasing | Minimum incre. I mux [Dec I minil inc. 1=-1, fr-1) = (-3)" - 26-13°+5 =) 1 - 275 (-1, 4) = Minimum 230, $10) = (0)" - 2(0)+5 = 5. (0-5) =) -Murimum I=1, #11 (1)" - 210²75 ET-2+5 =4 (1,437 Minimym