# Find min value of a complex Root function [closed]

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}]


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}}

• Many thanks! I tried findminium and min, but clearly I forgot there is a minimize. Jun 14 at 12:49
• Minimize results in a polynomial of degree 35. NMinimize is preferable here. Jun 14 at 14:17

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}}  *)