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Question-188270




Question Number 188270 by mnjuly1970 last updated on 27/Feb/23
Commented by mnjuly1970 last updated on 27/Feb/23
  in  AB^Δ C prove :      (1/(cos(A))) +(1/(cos(B))) +(1/(cos(C))) ≥ 6          note : 0 < A , B , C < 90^°
$$\:\:{in}\:\:{A}\overset{\Delta} {{B}C}\:{prove}\::\: \\ $$$$\:\:\:\frac{\mathrm{1}}{{cos}\left({A}\right)}\:+\frac{\mathrm{1}}{{cos}\left({B}\right)}\:+\frac{\mathrm{1}}{{cos}\left({C}\right)}\:\geqslant\:\mathrm{6} \\ $$$$\: \\ $$$$\:\:\:\:\:{note}\::\:\mathrm{0}\:<\:{A}\:,\:{B}\:,\:{C}\:<\:\mathrm{90}^{°} \\ $$$$ \\ $$
Answered by mahdipoor last updated on 27/Feb/23
f(x,y,z)=(1/(cos(x)))+(1/(cos(y)))+(1/(cos(z)))  g(x,y,z)=x+y+z=180   (eq I)  p=(a,b,c) ,  f(p)=min f  ⇒▽f^→ (p)=λ▽g^→ (p)  ⇒(−((sin(a))/(cos^2 (a))))i+(−((sin(b))/(cos^2 (b))))j+(−((sin(c))/(cos^2 (c))))k=  =λ(i+j+k)    (eq II)  ⇒⇒eq I  & II ⇒  ((sina)/(cos^2 a))=((sinb)/(cos^2 b))=((sinc)/(cos^2 c))=−λ  a+b+c=180  ⇒⇒a=b=c=60⇒min f=(3/(cos(60)))=6
$${f}\left({x},{y},{z}\right)=\frac{\mathrm{1}}{{cos}\left({x}\right)}+\frac{\mathrm{1}}{{cos}\left({y}\right)}+\frac{\mathrm{1}}{{cos}\left({z}\right)} \\ $$$${g}\left({x},{y},{z}\right)={x}+{y}+{z}=\mathrm{180}\:\:\:\left({eq}\:{I}\right) \\ $$$${p}=\left({a},{b},{c}\right)\:,\:\:{f}\left({p}\right)={min}\:{f} \\ $$$$\Rightarrow\bigtriangledown\overset{\rightarrow} {{f}}\left({p}\right)=\lambda\bigtriangledown\overset{\rightarrow} {{g}}\left({p}\right) \\ $$$$\Rightarrow\left(−\frac{{sin}\left({a}\right)}{{cos}^{\mathrm{2}} \left({a}\right)}\right){i}+\left(−\frac{{sin}\left({b}\right)}{{cos}^{\mathrm{2}} \left({b}\right)}\right){j}+\left(−\frac{{sin}\left({c}\right)}{{cos}^{\mathrm{2}} \left({c}\right)}\right){k}= \\ $$$$=\lambda\left({i}+{j}+{k}\right)\:\:\:\:\left({eq}\:{II}\right) \\ $$$$\Rightarrow\Rightarrow{eq}\:{I}\:\:\&\:{II}\:\Rightarrow \\ $$$$\frac{{sina}}{{cos}^{\mathrm{2}} {a}}=\frac{{sinb}}{{cos}^{\mathrm{2}} {b}}=\frac{{sinc}}{{cos}^{\mathrm{2}} {c}}=−\lambda \\ $$$${a}+{b}+{c}=\mathrm{180} \\ $$$$\Rightarrow\Rightarrow{a}={b}={c}=\mathrm{60}\Rightarrow{min}\:{f}=\frac{\mathrm{3}}{{cos}\left(\mathrm{60}\right)}=\mathrm{6} \\ $$

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