Question

Find the intervals from which f is increasing or decreasing for f(x)=3x^2/5-x^3/5

Answer 1

`f(x)=3x^(2/5)-x^(3/5)`

We get the derivative :
`f'(x)=3*2/5*x^(2/5-1)-3/5*x^(3/5-1)=(6)/(5x^(3/5))-(3)/(5x^(2/5))`

f'(x) is negative for x<0.

Let's look at the critical points other than at x=0:
f'(x)=0 <=> 6/5-(3/5)x^(1/5)=0 <=> x^(1/5)=(6/5)*(5/3)=2 hence x=2^5=32

There is a critical point at x=32; for x<=32 f' is positive, for x>=30 f' is negative.

The derivative is positive for positive xs and negative for negative xs, hence f is increasing for 32>=x>0, and decreasing for x<0 and x>32