Question

For [-12,12] f(x)=x^5(x+3)^4 on which interval is the function increasing?

Answer 1

f(x)=x^5(x+3)^4
f'(x) = 5x^4 (x+3)^4 + 4(x+3)^3 x^5
=x^4 (x+3)^3 (5(x+3) +4x)
=x^4 (x+3)^3 (9x +15)
=3(x^4)(x+3)^3 (3x+5)
to check if f(x) is increasing or decreasing, we have to find out the turning points first. How to find it?
Since turning points have slope =0, set f'(x)=0
3(x^4)(x+3)^3 (3x+5) =0
x=0, -3 or -5/3
then, we can get ranges : -3<x<-5/3 or -5/3<x<0 or x>0
for -3<x<-5/3, put any values in this range to f'(x). For example, we sub. x=-2 into f'(x).Then, we can get f'(x)<0.
Similarly,for -5/3<x<0 , f'(x) >0
for x>0, f'(x) >0

Note that f'(x)>0 means that f(x) is strictly increasing
f'(x)<0 means that f(x) is strictly decreasing
Then we now can figure out that for -5/3<x<0 and 0<x<12, f(x) is increasing

Answer 2

THANKS