Thread: Very confused...(derivatives)

I am working on some work that was assigned to me in my calculus class and I don't understand how I am supposed to start this problem. Can anyone guide me in the right direction or get me started off?

Here's a link to the problem: ImageShack® - Online Photo and Video Hosting

Not asking for anyone to do it for me, I just need some help getting started .

Thanks

let a point on the parabola's upper branch be ... note that symmetry will take care of the normal line on the lower branch

slope between and ...



slope of the upper branch at any point ...

slope normal to any point ...

note that the normal slope equals the slope between and and that

set up an equation, solve for in terms of and then sub into the inequality

I worked on this for a while last night and came up with this for an answer...have I answered it correctly?


3a. Let c > 0 be a point along the positive real axis. Then the tangent to the parabola at (c,sqrt(c)) has slope
1/(2sqrt(c)) => the normal line has slope -2sqrt(c). point slope form: the normal line has equation
y - sqrt(c) = -2sqrt(c)(x - c)

this intercepts the x-axis where y = 0 => -sqrt(c) = -2sqrt(c)(a-c) => a = 1/2 + c.

therefore a must be at least 1/2.

3b. the normal lines are perpendicular when they have pi/4 angles to the real axis.
this occurs when -2sqrt(c) = -1 => c = 1/4 => a = 3/4.

That looks good to me.