How do you find all intervals where the function #f(x)=1/3x^3+3/2x^2+2# is increasing?

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Answer 1 · 2014-09-25T20:29:46

#f# is increasing on #(-infty,-3]# and #[0,infty)#.

The graph of #f# looks like this:

enter image source here

Let us look at some details.

#f(x)=1/3x^3+3/2x^2+2#

By solving #f'(x)=0# for #x#,

#f'(x)=x^2+3x=x(x+3)=0#,

we find the critical values: #x=-3, 0#.

Using the critical values to divide #(-infty,infty)# into

#(-infty,-3]#, #[-3,0]#, and #[0,infty)#

and we choose sample values #-4#, #-2# and #1# for the intervals above, respectively. (We can use any number on each interval excluding endpoints.)

#f'(-4)=4>0 Rightarrow f# is increasing on #(-infty,-3]#

#f'(-2)=-2<0 Rightarrow f# is decreasing on #[-3,0]#

#f'(1)=4>0 Rightarrow f# is increasing on #[0,infty)#