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Could someone tell me how to obtain the min value of this function when lambda is between 0 and 1:

Root[λ^2 + 21 λ^3 + 381 λ^4 + 
   247 λ^5 + 55 λ^6 - 
   3 λ^7 + (-8192 - 40960 λ - 81280 λ^2 - 
      79872 λ^3 - 38620 λ^4 - 7104 λ^5 + 
      148 λ^6 - 8 λ^7) #1 + (65536 + 
      262144 λ + 368640 λ^2 + 229261 λ^3 + 
      20176 λ^4 - 10016 λ^5 + 268 λ^6 - 
      4 λ^7) #1^2 + (-98304 λ - 112640 λ^2 + 
      21528 λ^3 + 26776 λ^4 - 3556 λ^5 + 
      96 λ^6) #1^3 &, 3]

I plot the graph using

Plot[Root[λ^2 + 21 λ^3 + 381 λ^4 + 
    247 λ^5 + 55 λ^6 - 
    3 λ^7 + (-8192 - 40960 λ - 81280 λ^2 - 
       79872 λ^3 - 38620 λ^4 - 7104 λ^5 + 
       148 λ^6 - 8 λ^7) #1 + (65536 + 
       262144 λ + 368640 λ^2 + 229261 λ^3 + 
       20176 λ^4 - 10016 λ^5 + 268 λ^6 - 
       4 λ^7) #1^2 + (-98304 λ - 112640 λ^2 + 
       21528 λ^3 + 26776 λ^4 - 3556 λ^5 + 
       96 λ^6) #1^3 &, 3], {λ, 0, 1}, 
 PlotRange -> {0, 12}]

The output

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2 Answers 2

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expr=Root[λ^2 + 21 λ^3 + 381 λ^4 + 
    247 λ^5 + 55 λ^6 - 
    3 λ^7 + (-8192 - 40960 λ - 81280 λ^2 - 
       79872 λ^3 - 38620 λ^4 - 7104 λ^5 + 
       148 λ^6 - 8 λ^7) #1 + (65536 + 
       262144 λ + 368640 λ^2 + 229261 λ^3 + 
       20176 λ^4 - 10016 λ^5 + 268 λ^6 - 
       4 λ^7) #1^2 + (-98304 λ - 112640 λ^2 + 
       21528 λ^3 + 26776 λ^4 - 3556 λ^5 + 
       96 λ^6) #1^3 &, 3];
Minimize[{expr,0 < λ < 1}, λ]
% // N

{4.15495, {λ -> 0.497444}}

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2
  • $\begingroup$ Many thanks! I tried findminium and min, but clearly I forgot there is a minimize. $\endgroup$
    – Jethro
    Jun 14 at 12:49
  • 1
    $\begingroup$ Minimize results in a polynomial of degree 35. NMinimize is preferable here. $\endgroup$
    – user64494
    Jun 14 at 14:17
2
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In response to the OP's comment:

expr = Root[λ^2 + 21 λ^3 + 381 λ^4 + 
     247 λ^5 + 55 λ^6 - 
     3 λ^7 + (-8192 - 40960 λ - 81280 λ^2 - 
        79872 λ^3 - 38620 λ^4 - 7104 λ^5 + 
        148 λ^6 - 8 λ^7) #1 + (65536 + 
        262144 λ + 368640 λ^2 + 229261 λ^3 + 
        20176 λ^4 - 10016 λ^5 + 268 λ^6 - 
        4 λ^7) #1^2 + (-98304 λ - 
        112640 λ^2 + 21528 λ^3 + 26776 λ^4 - 
        3556 λ^5 + 96 λ^6) #1^3 &, 3];
FindMinimum[expr, {λ, 1}]

(*  {4.15495, {λ -> 0.497444}}  *)
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