What is the upper envelope of the family of ballistic curves?

[mathmathmad]

Homework Statement

find the upper envelope for the family of ballistic curves :

y = ax - [x^2(a^2+1)]/2

an upper envelope is a curve y=F(x) such that for each x fixed, F(x) is the maximum of g(a) = ax - [x^2(a^2+1)]/2 for a in R

Homework Equations

The Attempt at a Solution

diff g(a) wrt to a and equate to 0 and get a=1/x so F(x) is 1/x?? :(

 

[ystael]

You are right to start by finding the maximum of g_x(a) = ax - \frac12 x^2 (a^2 + 1), but you need to prove that the maximum of g_x occurs at the single critical point you find.

However, you have found the value of a which maximizes g_x(a) = F(x) at that x --- not what you want, which is the value of the function at that point, expressed as a function of x. What further computation is necessary here?

 

[blackscorpion]

Am I right in thinking you then have to sub the a=1/x back into the original y to get,

y=F(x)=(1/2) x^2

and mathmathmad your a warwick student right?

 

[vintwc]

Am I right in thinking you then have to sub the a=1/x back into the original y to get,

y=F(x)=(1/2) x^2

and mathmathmad your a warwick student right?

I do not think so as it is not related with F(x) being maximum of g(a). Btw blackscorpian, I've sent you a pm.

 

[mathmathmad]

is it F(x)=1/2 - x^2/2 instead?

@blackscorpion : yes... I am. you too?

 

[vintwc]

Yes it is

 

[blackscorpion]

Ahhh crap, it is aswell.
Stupidly canceled the ones forgettin bout the over 2 part.
Well that's a mark thrown away, lol