Finding Critical Numbers for a Polynomial Function with Power and Chain Rules

[frosty8688]

1. Find the critical numbers of F(x) = x^{\frac{4}{5}}(x-4)^{2}



2. Power rule then chain rule



3. F'(x) = \frac{4}{5}x^{\frac{-1}{5}} (x-4)^{2}*2(x-4) I know two critical numbers are 0 and 4 and I am having problems finding the third one.

 

[Mark44]

1. Find the critical numbers of F(x) = x^{\frac{4}{5}}(x-4)^{2}
2. Power rule then chain rule

and product rule?

3. F'(x) = \frac{4}{5}x^{\frac{-1}{5}} (x-4)^{2}*2(x-4)

The first rule to use would be the product rule. It doesn't look to me like you used that rule.

I know two critical numbers are 0 and 4 and I am having problems finding the third one.

 

[HallsofIvy]

Another way to do this is to write
F(x)= x^{\frac{4}{5}}(x^2- 8x+ 16)= x^{\frac{14}{5}}- 8x^{\frac{9}{5}}+ 16x^{\frac{4}{5}}

 

[frosty8688]

That makes it easier to understand.

 

[Mark44]

Yes, but it would also be useful to use the product rule (correctly). The results should be the same for either method, but you might need to use some algebra to confirm that they are the same.