Determine the intervals on which the function is concave up or concave down. (En
ID: 2858961 • Letter: D
Question
Determine the intervals on which the function is concave up or concave down. (Enter your answers using interval notation. Enter EMPTY or for the empty set.) f(x) = 4x/(4x)^2 + 27 Find the critical points of the function and determine whether the critical point is a local minimum or maximum (or neither). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x) = (x - 8)^3/5 local minimum local maximum Find the points of inflection of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) Determine the intervals on which the function is concave up or concave down. (Enter your answers using interval notation. Enter EMPTY or for the empty set.) concave up concave downExplanation / Answer
48) f(x)= 4x/[(4x)2+27]
f(x)= 4x/[16x2+27]
differentiate with respect to x on both sides. quotient rule:(u/v)'=(u'v -uv')/v2,u=4x,v=16x2+27
f '(x)= [4(16x2+27)- 4x(16*2x +0)]/[16x2+27]2
f '(x)= [64x2+108- 128x2 -0]/[16x2+27]2
f '(x)= [108- 64x2]/[16x2+27]2
f '(x)= 4[27- 16x2]/[16x2+27]2
differentiate with respect to x on both sides
f ''(x)= 4[(0- 16*2x)(16x2+27)2 -(27- 16x2)2(16x2+27)(16*2x +0)]/[[16x2+27]2]2
f ''(x)= 4[(- 32x)(16x2+27)2 -(27- 16x2)2(16x2+27)(32x)]/[16x2+27]4
f ''(x)= -4(16x2+27)(32x)[(16x2+27) +2(27- 16x2)]/[16x2+27]4
f ''(x)= -4(16x2+27)(32x)[16x2+27+54- 32x2)]/[16x2+27]4
f ''(x)= -4(16x2+27)(32x)[81- 16x2]/[16x2+27]4
fuction has inflection point when f "(x)=0
-4(16x2+27)(32x)[81- 16x2]/[16x2+27]4=0
-4(16x2+27)(32x)[81- 16x2]=0
(32x)[81- 16x2]=0
x=0,81- 16x2=0
x=0,x2=81/16
x=0, x =-9/4, x =9/4
x=-9/4, x =0, x =9/4
when x<-9/4, f "(x)<0
-9/4<x<0, f "(x)>0
0<x<9/4, f "(x)<0
x>9/4, f "(x)>0
function is concave up when f "(x)>0
=>(-9/4,0)U(9/4,infinity)
function is concave down when f "(x)<0
=>(-infinity,-9/4)U(0,9/4)
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