# Upper envelope of a function

1. Feb 23, 2010

1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. 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?? :(

2. Feb 24, 2010

### 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?

3. Feb 24, 2010

### 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

4. Feb 24, 2010

### vintwc

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.

5. Feb 25, 2010

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

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

6. Feb 25, 2010

### vintwc

Yes it is

7. Feb 25, 2010

### blackscorpion

Ahhh crap, it is aswell.
Stupidly cancelled the ones forgettin bout the over 2 part.
Well thats a mark thrown away, lol