Maximum gradient of a normal to the curve

[abdo799]

Homework Statement

complete problem attached

Homework Equations

The Attempt at a Solution

part I in this question was a bit tricky but i managed to solve it , when i read part II i understood nothing , he usually asks about the tangent not the normal , he asks about the point where the gradient of the normal is maximum and i have no idea how to get this , when i read the answers he said we should differentiate again then = it to 0 to find x , why did this work?

 

[abitslow]

The tangent lines intersecting the given equation form a family, one for each point. Right?
The normal line is simply obtained from any straight line, right?
That makes the set of normals a family of lines intersecting the given eqn at every point on it.
From there, it is a simple application of the definition of gradient. Oh, and obviously of finding the stationary points of that family. Not sure what else to tell you. It seems to be straighforward algebra as long as you understand what the various things are you are dealing with (and can differentiate simple functions).

 

[abdo799]

i got it...thanks