This is a calculus homework question but in reality the problem here is understanding the algebra.

Question

Find the Critical Numbers of f(x) = x^(3/5)(4-x)

The product rule gives -x^(3/5) + (12-3x)/(5x^(2/5)

My booklet explains that simplifying this gives you 12-8x / (5x^(2/5).

How did it get to that?

Answer 1

first = x^(3/5)
second = (4-x)

derivative = first derivative of second + second derivative of first

first derivative of second = x^(3/5)*-1

second derivative of first
= (4-x)[ (3/5){x^-(2/5)}
= (1/5)(12 -3x)x^-(2/5)
so
-x^(3/5) + (1/5)(12 -3x)x^-(2/5)

but x^(3/5) = x^(-2/5)x^(5/5)
that is what you missed

Answer 2

Thanks!

Answer 3

You are welcome.