Why Does f'(x) Have Only 3 Critical Numbers on (0,10)?

[Hothot330]

Homework Statement

The first derivative of the function f is given by f '(x)= (((cos^2)x)/(x)) - 1/5
How many critical numbers does f have on the open interval (0,10)?

Homework Equations

The Attempt at a Solution

I already got this question wrong but I don't know why I got it wrong. The answer is 3 but when I graph it I see 6 critical numbers. So why is it 3 and not 6? Please explain.

 

[rocomath]

Let's clear up any confusion about what the problem is ...

f'(x)=\frac{\cos^2 x}{x}-\frac 1 5

Correct?

 

[Hothot330]

Correct

 

[Hothot330]

I tried taking the derivative of the derivative and graphed that but it still gives me an image of 6 critical points.

 

[HallsofIvy]

I don't understand why that would happen. A critical point occurs where the derivative is 0 or does not exist. Clearly your derivative does not exist at x= 0. To determine where f'= 0, I graphed y= 5cos²(x) and y= x. They cross in 3 points.

Wait, did you differentiate again? You said what you gave was f '. To determine where f has critical points, you should be graphing that, not its derivative.