θ+φ+ψ=π (angles of a △) find maximum of (φ−θ)^2 +(ψ−φ)^2 +(ψ−θ)^2 .


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Question Number 141269 by ajfour last updated on 17/May/21

$θ + ϕ + ψ = π (a n g l e s o f a △)$ $f i n d m a x i m u m o f$ ${(ϕ - θ)}^{2} + {(ψ - ϕ)}^{2} + {(ψ - θ)}^{2} .$
Commented by MJS_new last updated on 17/May/21

$maximum 2 π^{2} when two angles are zero and$ $the third one is π$
Commented by ajfour last updated on 17/May/21

$t h a n k s s i r, a n y a n a l y t i c a l$ $m e t h o d . . ?$
Commented by MJS_new last updated on 17/May/21

$sorry for using a, b, c instead of your variable$ $names, I^{'} m in a hurry$ $(1)$ $c = π - a - b$ $\Rightarrow$ $6 a^{2} + 6 a b + 6 b^{2} - 6 π a - 6 π b + 2 π^{2} \to \max$ $\frac{d}{d b} [. . .] = 6 a + 12 b - 6 π = 0 \Rightarrow b = \frac{π - a}{2}$ $but this leads to a = b = c = \frac{π}{3} and a minimum$ $\Rightarrow$ $we have to search along the borders$ $0 ⩽ a, b ⩽ π$ $b = 0$ $6 a^{2} - 6 π a + 2 π^{2} \to \max$ $again only a minimum \Rightarrow$ $a = 0 \lor a = π \Rightarrow \max = 2 π^{2}$
	Answered by TheSupreme last updated on 17/May/21
	$Γ(∅,θ)=f(∅,θ,π−∅−θ)=(∅−θ)^2 +(π−2∅−θ)^2 +(π−∅−2θ)^2 grad Γ= { ((2(∅−θ)−4(π−2∅−θ)−2(π−∅−2θ)=0)),((−2(∅−θ)−2(π−2∅−θ)−4(π−∅−2θ)=0)) :} { ((2∅+θ=π)),((∅+2θ=π)) :} (∅,θ)=((π/3),(π/3)) Γ((π/3),(π/3))=0 (absolute min) set ∅=kθ Λ(θ,k)=f(kθ,θ,π−kθ−θ)=θ^2 (k−1)^2 +(π−2kθ−θ)^2 +(π−kθ−2θ)^2 θ^2 [(k−1)^2 +(2k+1)^2 +(k+2)^2 ]−2θπ[(2k+1)+(k+2)]+2π^2 as you can see f is not limited f has no max$
	$Γ (\emptyset, θ) = f (\emptyset, θ, π - \emptyset - θ) = {(\emptyset - θ)}^{2} + {(π - 2 \emptyset - θ)}^{2} + {(π - \emptyset - 2 θ)}^{2}$ $g r a d Γ = {\begin{cases} 2 (\emptyset - θ) - 4 (π - 2 \emptyset - θ) - 2 (π - \emptyset - 2 θ) = 0 \\ - 2 (\emptyset - θ) - 2 (π - 2 \emptyset - θ) - 4 (π - \emptyset - 2 θ) = 0 \end{cases}$ ${\begin{cases} 2 \emptyset + θ = π \\ \emptyset + 2 θ = π \end{cases}$ $(\emptyset, θ) = (\frac{π}{3}, \frac{π}{3})$ $Γ (\frac{π}{3}, \frac{π}{3}) = 0 (a b s o l u t e m i n)$ $s e t \emptyset = k θ$ $Λ (θ, k) = f (k θ, θ, π - k θ - θ) = θ^{2} {(k - 1)}^{2} + {(π - 2 k θ - θ)}^{2} + {(π - k θ - 2 θ)}^{2}$ $θ^{2} [{(k - 1)}^{2} + {(2 k + 1)}^{2} + {(k + 2)}^{2}] - 2 θ π [(2 k + 1) + (k + 2)] + 2 π^{2}$ $a s y o u c a n s e e f i s n o t l i m i t e d$ $f h a s n o m a x$
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