How do you find intercepts, extrema, points of inflections, asymptotes and graph #y=3x^4+4x^3#?

1 Answer
Mar 14, 2017

See explanation below

Explanation:

#y=3x^4+4x^3#

This is a polynomial function

The domain of #y# is #D_y=RR#

We calculate the first and second derivatives

#dy/dx=12x^3+12x^2#

The critical points are when

#dy/dx=0#

#12x^3+12x^2=0#

#12x^2(x+1)=0#

Therefore,

#x=0# and #x=-1#

When #x=0#, #=>#, #y''=0#, #=>#, point of inflexion

When #x=-1#, #=>#, #y''>0#, a minimum

We build a sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-1##color(white)(aaaa)##0##color(white)(aaaaa)##+oo#

#color(white)(aaaa)##x+1##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#

#color(white)(aaaa)##y##color(white)(aaaaaaaaa)##↘##color(white)(aaaa)##↗##color(white)(aaaa)##↗#

We calculate the second derivative

#(d^2y)/dx^2=36x^2+24x#

#(d^2y)/dx^2=0#

#36x^2+24x=0#

#12x(3x+2)=0#

Therefore,

#x=0# and #x=-2/3#

These are the points of inflexions

We construct a chart

#color(white)(aaaa)## Interval##color(white)(aaaa)##]-oo,-2/3[##color(white)(aaaa)##]-2/3,0[##color(white)(aaaa)##]0,+oo[#

#color(white)(aaaa)## sign( y'')##color(white)(aaaaaaaaa)##+##color(white)(aaaaaaaaaa)##-##color(white)(aaaaaaaa)##+#

#color(white)(aaaa)##y##color(white)(aaaaaaaaaaaaaaaaa)##uu##color(white)(aaaaaaaaaa)##nn##color(white)(aaaaaaaa)##uu#

The #color(red)( Intercepts)# are

#(0,0)# and

#3x^4+4x^3=0#

#x^3(3x+4)=0#

#x=0# and #x=-4/3#

#color(red)("limits")#

#lim_(x->+-oo)y=lim_(x->+-oo)3x^4=+oo#

graph{3x^4+4x^3 [-3.082, 3.075, -1.538, 1.542]}