How do you use the first derivative test to determine the intervals on which #f(x)=x^4+3x^3+3x^2+1# is increasing or decreasing and whether each critical point is a local maximum, minimum or neither?

1 Answer
Sep 24, 2015

Answer:

See the explanation.

Explanation:

Using the rules:
#y=x^n, y'=nx^(n-1)#

and

#h(x)=f_1(x)+f_2(x)+...+f_n(x)#
#h'(x)=f'_1(x)+f'_2(x)+...+f'_n(x)#

we get:

#f'(x)=4x^3+9x^2+6x=x(4x^2+9x+6)#

Lets examine expression #4x^2+9x+6#:
#4x^2+9x+6=0#
#D=b^2-4ac=9^2-4*4*6=81-96=-15#
#D<0# which means that quadratic function #4x^2+9x+6# doesn't have real zeros. Furthermore, #a=4>0# (coefficient in front of the x^2), so the function is concave.

So, expression #4x^2+9x+6>0# for #AAx in R#.

#f'(x)=0 <=> x(4x^2+9x+6)=0 <=> x=0#

#AAx<0: f'(x)<0# and #f# is decreasing
#AAx>0: f'(x)>0# and #f# is increasing

#f'(x)# changes sign in #x=0# and #f(x)# has minimum value #f_min=f(0)=1#