finding the critical points of a function. I am a little stumped after I take the first dereivative.

Question

If U and V are positive constants, find the critical point of the function.

F(t) = Ue^t + Ve^-t

My work:

F'(t) = Ue^t + -Ve^-t

0= Ue^t - Ve^-t

after this I have no idea how to find the critical points. If I factor out e I am left with constants and e expressions. e functions cannot =0 ever and the constants have no variable attached so I have no clue what to solve for. Please help me figure out where I am thinking about this function in the wrong way.

Answer 1

You were proceeding correctly.  Note that the second derivative is Ue^t + Ve^-t is greater than 0 for all t because U and V are positive.  Thus, the function is strictly convex, so if the first derivative equals 0, this is a minimum.

Then, Ue^t - Ve-^t = 0, so Ue^t = Ve-^t   Multiplying through by e^t / U, we get e^2t = V/U.  Taking the natural log on both sides,

2t = ln(V/U)

t = 1/2 ln(V/U)

Answer 2

Factor out e^-t.

 

e^-t(Ue^2t - V) = 0

 

Set the term in parenthesis equal to zero.

 

Ue^2t - V = 0

 

Ue^2t = V

 

2tln(Ue) = ln(V)

 

2t = ln(V) / ln(Ue)

 

2t = ln(V) / (lnU + 1)

 

t = (1/2)[ln(V) / (lnU + 1)]