Parametric Equations finding largest radius

[Painguy]

Homework Statement

Suppose that r = f (θ) defines a polar graph. Find an expression for dx/dθ. It should not involve the letter r. Explain a procedure to determine the farthest that the graph r = f (θ) extends to the left and to the right (Hint: If x = x0 is the x - value of the point that is farthest left, how does x0 compare with all other x-values?).

Homework Equations

The Attempt at a Solution

The wording of the question seems a bit off especially the hint "is the x - value of the point that is farthest left".
Here is how I started the problem. I know that x=r*cos(θ) so i subsituded r for f(θ) to get
x=f(θ)cos(θ). My guess is that when the derivative of dx/dθ is equal to 0 then that means that the polar graph is turning back around so I take the derivative of x to get dx/dθ =-f(θ)sin(θ) +f'(θ)cos(θ)
then I set it equal to 0 to get
-f(θ)sin(θ) +f'(θ)cos(θ)=0
f(θ)sin(θ) =f'(θ)cos(θ)
tan(θ)=f'(θ)/f(θ)
tan(θ)=f'(θ)/(x/cos(θ))
x=(f'(θ) tan(θ)cos(θ))

I don't really know what I'm doing at this point. How should I approach this problem?

 

[haruspex]

tan(θ)=f'(θ)/f(θ)

It all looks fine to here. In practice, you would know f(), so could then solve the above equation to find the theta of interest, then compute x from x = f(θ) cos(θ). There's no point in bringing x back into the equation first unless you can eliminate theta and get an equation in x only - which you can't without knowing what f is.

 

[Painguy]

It all looks fine to here. In practice, you would know f(), so could then solve the above equation to find the theta of interest, then compute x from x = f(θ) cos(θ). There's no point in bringing x back into the equation first unless you can eliminate theta and get an equation in x only - which you can't without knowing what f is.

I see. That makes sense. I guess I was expecting a bit more out of the problem. Is there any point to the hint that was provided? I feel like I'm still missing a big chunk of the question.

 

[haruspex]

I see. That makes sense. I guess I was expecting a bit more out of the problem. Is there any point to the hint that was provided? I feel like I'm still missing a big chunk of the question.

You used the hint when you looked for the extremum of x. One thing you have not done is show how to determine it's a leftmost value, not a rightmost one.