Question

Consider the function below. h(x) = 7x5 - 12x3 + x (a) Find the intervals of increase. (Enter your answer using interval notation.) X Find the intervals of decrease. (Enter your answer using interval notation.) (b) Find the local maximum value(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) -1,0.169 X Find the local minimum value(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) (c) Find the inflection points. (x, y) = (smallest x-value) (x, y) = (x, y) = (largest x-value) Find the intervals the function is concave up. (Enter your answer using interval notation.) Find the intervals the function is concave down. (Enter your answer using interval notation.) (d) Use the information from parts (a)-(c) to sketch the graph of the function. Check your work with a graphing device if you have one. (Do this on paper. Your instructor may ask you to turn in this work.)

Answer 1

Solution or hla)= 728_1223+2 c) differentiate wisstou de weget. h(a)= 35x"_36x² + 1 hem = 3574-132-352+ + 1 ² [35²²-1] -1 [352²1 hm) -62-1) a 3522-17 So, critical points are 2- El ; I th hima of I v35 35 h (n)= 35x4-36x²+1 y el لا بنا 스 35 35 So, 8 62) = 35 (16) - 3644 +1 = 560-144+1 = 4470 8'1-0.5)= 3560.5)4- 3660-5)+1 - 35x10-0625) - 36 (0-25)+1 -2.1875 9 tl 5-5.812520 *(0)=0-0+1=130 8'40.5) 35 (+0.5)4-3660.5)*+1 -5.8125co So, Interval for of Increasing 60,-1) 01- Ulice Intowal of decreasing (-1,- --ve) u (V) 1' (2) = 351162-36444 -44770

b) for minina of a function + + + + -V35 V35 35 35 Of Sigis of f'(n2 changes from 0 to ② =) local minimate point & vice versa fose local marina pt. are local minimal pt. f(x)=725 - 1223+ 8(1): 7-12+1= -4 & flag J-71-9839-rele (9 2 35 35 &n=1 11635 6125 So, C.6526 ) & (1-4) are local minima pts. Similarly (-1,4) & (133/2, 116 s) are pts of local maxima pts. c) h" (n) =24022-722 put h "(n)=0 14043-722=0 » [14024-72)-0 2=0 & x²= 36 70 So n=0 & 2 = 7 3 V70 35 18 35 a = + 352 are inflection points

for (n= for the f100=0 3 V70 35 64.3 35 35 f(-320) = +2.49570 35 875 for a=+3770 35 3/70 = 7/3/70 35 35 2242570 35 So, are 875 87 7 (-37,249 17p ), (050) & A39 -249 V2 inflection pt. for concavity h"(x) = 140x3 722 h"(-1)=140(-1)+72 Concave Concure Concave Up Con cave dorin up dow o 130 35 35 -68 co h" foil) - 140 60.1)?-7260.1) = -0.14 + 7.2 = 706 70 So, Concave downs (700,- 3870 3570 35 35 Concave upward, (-3470, o) u (3570 35 h" (0:1) 0.14 -7 2 5-7.06 <0 35 h"(1) = 140-79 = 6870

a) Interval of Increase :-6-00,- 19 U L-VBS, JBS Jul1,co) Interval of decrease :- (-1,- very) Uliving b) local marina at (-1,4) & ( 132 116 535 35? ) 6125 Local minima a C-3, -116835 P6125 c) Infketen pto (-3470, 249670 5).Co 0,0), 1317, 249. Jeg 249.vis) Concave up r (-3) US, 20 Concalie down ***37)U(0,31%)

_y_4 3 -2- -2 -3