Câu hỏi

Cho a, b, c là độ dài 3 cạnh của tam giác ABC. Tìm giá trị nhỏ nhất của

Cho a, b, c là độ dài 3 cạnh của tam giác ABC. Tìm giá trị nhỏ nhất của $\mathrm T=\left(\frac{\mathrm a}{{\mathrm m}_{\mathrm a}}\right)^\sqrt3+\left(\frac{\mathrm b}{{\mathrm m}_{\mathrm b}}\right)^\sqrt3+\left(\frac{\mathrm c}{{\mathrm m}_{\mathrm c}}\right)^\sqrt3$

R. Roboctvx57

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Giải thích

Bổ đề: m a a + m b b + m c c ≥ 2 3 ( 1 ) ( 1 ) ⇔ a . m a a 2 + b . m b b 2 + c . m c c 2 ≥ 2 3 Ta có: a . m a = a 4 2 b 2 + 2 c 2 − a 2 = 2 1 ( 2 b 2 + 2 c 2 − a 2 ) a 2 = 2 3 1 ( 2 b 2 + 2 c 2 − a 2 ) 3 a 2 ≤ ≤ 2 3 1 2 ( 2 b 2 + 2 c 2 − a 2 ) + 3 a 2 = 2 3 a 2 + b 2 + c 2 Suy ra: + ⎩ ⎨ ⎧ m a a = a . m a a 2 ≥ 2 3 1 ( a 2 + b 2 + c 2 ) a 2 = a 2 + b 2 + c 2 2 3 a 2 m b b = b . m b b 2 ≥ 2 3 1 ( a 2 + b 2 + c 2 ) b 2 = a 2 + b 2 + c 2 2 3 b 2 m c c = c . m c c 2 ≥ 2 3 1 ( a 2 + b 2 + c 2 ) c 2 = a 2 + b 2 + c 2 2 3 c 2 ⇒ m a a + m b b + m c c ≥ a 2 + b 2 + c 2 2 3 ( a 2 + b 2 + c 2 ) = 2 3 Sử dụng bất đẳng thức Bernoulli: Với a=b=c ⇔ △ ABC đều thì Min T =

Bổ đề: $\frac{a}{m _{a}} + \frac{b}{m _{b}} + \frac{c}{m _{c}} \geq 23 (1)$

$(1) \Leftrightarrow \frac{a ^{2}}{a . m _{a}} + \frac{b ^{2}}{b . m _{b}} + \frac{c ^{2}}{c . m _{c}} \geq 23$

Ta có:

$a . m_{a} = a \frac{2 b ^{2} + 2 c ^{2} - a ^{2}}{4} = \frac{1}{2} (2 b^{2} + 2 c^{2} - a^{2}) a^{2} = \frac{1}{2 3} (2 b^{2} + 2 c^{2} - a^{2}) 3 a^{2} \leq \leq \frac{1}{2 3} \frac{( 2 b ^{2} + 2 c ^{2} - a ^{2} ) + 3 a ^{2}}{2} = \frac{a ^{2} + b ^{2} + c ^{2}}{2 3}$

Suy ra:

$+ ⎩ ⎨ ⎧ \frac{a}{m _{a}} = \frac{a ^{2}}{a . m _{a}} \geq \frac{a ^{2}}{\frac{1}{2 3} ( a ^{2} + b ^{2} + c ^{2} )} = \frac{2 3 a ^{2}}{a ^{2} + b ^{2} + c ^{2}} \frac{b}{m _{b}} = \frac{b ^{2}}{b . m _{b}} \geq \frac{b ^{2}}{\frac{1}{2 3} ( a ^{2} + b ^{2} + c ^{2} )} = \frac{2 3 b ^{2}}{a ^{2} + b ^{2} + c ^{2}} \frac{c}{m _{c}} = \frac{c ^{2}}{c . m _{c}} \geq \frac{c ^{2}}{\frac{1}{2 3} ( a ^{2} + b ^{2} + c ^{2} )} = \frac{2 3 c ^{2}}{a ^{2} + b ^{2} + c ^{2}} \Rightarrow \frac{a}{m _{a}} + \frac{b}{m _{b}} + \frac{c}{m _{c}} \geq \frac{2 3 ( a ^{2} + b ^{2} + c ^{2} )}{a ^{2} + b ^{2} + c ^{2}} = 23$

Sử dụng bất đẳng thức Bernoulli:

$plus open curly brackets table attributes columnalign left end attributes row cell open parentheses fraction numerator square root of 3 straight a over denominator 2 straight m subscript straight a end fraction close parentheses to the power of square root of 3 end exponent plus square root of 3 minus 1 greater or equal than square root of 3 open parentheses fraction numerator square root of 3 straight a over denominator 2 straight m subscript straight a end fraction close parentheses end cell row cell open parentheses fraction numerator square root of 3 straight b over denominator 2 straight m subscript straight b end fraction close parentheses to the power of square root of 3 end exponent plus square root of 3 minus 1 greater or equal than square root of 3 open parentheses fraction numerator square root of 3 straight b over denominator 2 straight m subscript straight b end fraction close parentheses end cell row cell open parentheses fraction numerator square root of 3 straight c over denominator 2 straight m subscript straight c end fraction close parentheses to the power of square root of 3 end exponent plus square root of 3 minus 1 greater or equal than square root of 3 open parentheses fraction numerator square root of 3 straight c over denominator 2 straight m subscript straight c end fraction close parentheses end cell end table close rightwards double arrow open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses to the power of square root of 3 end exponent open square brackets open parentheses straight a over straight m subscript straight a close parentheses to the power of square root of 3 end exponent plus open parentheses straight b over straight m subscript straight b close parentheses to the power of square root of 3 end exponent plus open parentheses straight c over straight m subscript straight c close parentheses to the power of square root of 3 end exponent close square brackets plus plus 3 open parentheses square root of 3 minus 1 close parentheses greater or equal than 3 over 2 open parentheses straight a over straight m subscript straight a plus straight b over straight m subscript straight b plus straight c over straight m subscript straight c close parentheses greater or equal than 3 square root of 3 rightwards double arrow straight T equals open parentheses straight a over straight m subscript straight a close parentheses to the power of square root of 3 end exponent plus open parentheses straight b over straight m subscript straight b close parentheses to the power of square root of 3 end exponent plus open parentheses straight c over straight m subscript straight c close parentheses to the power of square root of 3 end exponent greater or equal than 3 open parentheses fraction numerator 2 over denominator square root of 3 end fraction close parentheses to the power of square root of 3 end exponent$

Với a=b=c $\Leftrightarrow △$ ABC đều thì Min T = $3 open parentheses fraction numerator 2 over denominator square root of 3 end fraction close parentheses to the power of square root of 3 end exponent$