What are the points of inflection of #f(x)= (x^2 - 8)/(x^2+3)
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Question

What are the points of inflection of #f(x)= (x^2 - 8)/(x^2+3)
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Charles Arocha · Accepted Answer

There is no point of inflexion of #f(x)# but there is a minimum point #(0,-8/3)#.

Some basic rules of differentiation are as follows, where #u# and #v# are functions of #x#: Addition / subtraction rule If #y=u+-v#, #dy/dx=d/dx(u)+-d/dx(v)# Chain rule #dy/dx=dy/(du)xx(du)/dx# Product rule If #y=uv#, #dy/dx=u(dv)/dx+v(du)/dx# Quotient rule If #y=u/v#, #dy/dx=(v(du)/dx-u(dv)/dx)/v^2# Let's get started, #f(x)=(x^2 - 8)/(x^2+3) # First, differentiate #f(x)#, #f'(x)=([d/dx(x^2-8)] (x^2+3)-[d/dx(x^2+3)] (x^2-8))/(x^2+3)^2# ( Quotient rule ) #color(white)(f'(x))=((2x) (x^2+3)-(2x) (x^2-8))/(x^2+3)^2# #color(white)(f'(x))=((2x) (cancel(x^2)+3cancel(-x^2)+8))/(x^2+3)^2# #color(white)(f'(x))=((2x)(11))/(x^2+3)^2# #color(white)(f'(x))=(22x)/(x^2+3)^2# Since, it is an stationary point, #d/dx[f(x)]=0#, #(22x)/(x^2+3)^2=0# #color(white)(xx/xxx.)22x=0# #color(white)(xxxxxx)x=0# Find the #y#-coordinate of the stationary point by substituting #x=0# into #f(x)#, #y= ((0)^2 - 8)/((0)^2+3)# #y=-8/3# Hence, the co-ordinate of the stationary point is #(0,-8/3)#. To find the nature of the stationary point, differentiate #f'(x)#, #f''(x)=([d/dx(22x)] (x^2+3)^2-[d/dx(x^2+3)^2] (22x))/(x^2+3)^4# ( Quotient rule ) #color(white)(f''(x))=((22)(x^2+3)^2-(4x)(x^2+3)(22x))/(x^2+3)^4# ( Product rule ) #color(white)(f''(x))=((22)(x^2+3)[(x^2+3)-(4x)(x)])/(x^2+3)^4# #color(white)(f''(x))=((22)(3-3x))/(x^2+3)^3# #color(white)(f''(x))=(66-66x^2)/(x^2+3)^3# Substitute #x=0# into #f''(x)#, #f''(x)=(66-66(0)^2)/((0)^2+3)^3# #color(white)(f''(x))=66/27 > 0# ( minimum point ) Hence, the stationary point #(0,-8/3)# is, in fact, a minimum point and not an inflexion point. Check: graph{(x^2 - 8)/(x^2+3) [-10, 10, -5, 5]}

Aurora Alvarado · Answer

undefined To find the points of inflection of $ f(x) = \frac{x^2 - 8}{x^2+3} $, we first find the second derivative, then set it equal to zero and solve for $ x $. After that, we can determine the corresponding $ y $ values.

What are the points of inflection of #f(x)= (x^2 - 8)/(x^2+3) #?