- #1

dyn

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- Homework Statement
- ##f(x) = 3x^{2/3}(5-x)##

Identify domain of ##f(x)## and is it odd or even or neither ?

Find the critical points of f(x)

- Relevant Equations
- Even function f(x) = f(-x) , odd function f(x) = -f(-x)

Critical points have the 1st derivative equal to zero

I have ##3x^{2/3}## as an even function although there is some debate as to this in another thread I started but the (5-x) factor means the function is neither odd or even. I also see the domain as all real numbers. Hopefully this is right ?

To find the critical points I differentiate f(x) to get ##10x^{-1/3}-5x^{2/3}## and set this equal to zero to get the critical points.

I can get the critical point of x=2 from this. The answer also states that x=0 is a critical point. This is the bit that confuses me.

As it stands ##10x^{-1/3}-5x^{2/3}## is not defined at x=0 but if I rearrange it as ##5x(2x^{-4/3}-x^{-1/3})## I get both critical values of x=0 and x=2.

How can the same equation be defined and not defined at the same time ? Without knowing the answer how would I know to rearrange the equation to get x=0 as an answer ?

Thanks

To find the critical points I differentiate f(x) to get ##10x^{-1/3}-5x^{2/3}## and set this equal to zero to get the critical points.

I can get the critical point of x=2 from this. The answer also states that x=0 is a critical point. This is the bit that confuses me.

As it stands ##10x^{-1/3}-5x^{2/3}## is not defined at x=0 but if I rearrange it as ##5x(2x^{-4/3}-x^{-1/3})## I get both critical values of x=0 and x=2.

How can the same equation be defined and not defined at the same time ? Without knowing the answer how would I know to rearrange the equation to get x=0 as an answer ?

Thanks