#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]}
