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Can Mathematica determine the symbolic expression in terms of τ that minimizes f[τ] ?

u[τ_] := n*E^(-λ*τ)  
f1[τ_] := k*τ
f2[τ_] := a*u[τ]
f3[τ_] := m*τ^2
f[τ_] := f1[τ] + f2[τ] + f3[τ]

a E^(-λ τ) n + k τ + m τ^2

So far so good. But when I use Minimize, it just gives back my input.

Minimize[f[τ], τ, Reals]
Minimize[a E^(-λ τ) n + k τ + m τ^2, τ, Reals]
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Do you have any assumptions about lambda, a, m, n and k? For example, n > 0 etc. – Kamov Sergey Jan 21 '14 at 21:34
All of the constants must be real and greater than zero. – Steve Jan 21 '14 at 21:37
For good formatting practice see editing-help. This browser extension is also useful. – ybeltukov Jan 21 '14 at 21:49
I don't know how to solve it using Minimize, but the answer can be obtained from Solve[D[f[t], t] == 0, t]. f[t] has only one extremum and it is minimum. – Kamov Sergey Jan 21 '14 at 21:52
Bingo Kamov ! I verified your solution numerically and it is spot on. Once again Mathematica did not let me down. But it is interesting that Minimize didn't "know" to call Solve for help. Thank you for your solution. – Steve Jan 21 '14 at 22:12
up vote 1 down vote accepted

By calculating a derivative it is easy to see that you get an equation

    D[f[\[Tau]], \[Tau]] == 0

(* k - a E^(-\[Lambda] \[Tau]) n \[Lambda] + 2 m \[Tau] == 0  *)

It is non-polynomial and non-linear. Equations of that sort Mathematica cannot solve analytically. For this reason Minimize does not work. You could try it numerically.

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