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Nov 23,2024

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\( 2a - b \) ifadesinin alabileceği en büyük tam sayı değeri 14'tür.

Solución

Verilen eşitsizlikler: 1. \( 2 < a < 7 \) 2. \( -1 < b < 3 \) İfademiz: \( 2a - b \) Bu ifadeyi maksimize etmek için \( a \) ve \( b \) değerlerini uygun şekilde seçmeliyiz. Öncelikle \( 2a \) ifadesini maksimize edelim. \( a \) değerinin en büyük olduğu sınırda, yani \( a \) için en büyük değer olan 7'ye yaklaşmalıyız. Ancak \( a < 7 \) olduğu için \( a \) değerini 7'ye çok yakın bir değer alabiliriz. Örneğin, \( a = 6.999 \) alalım. Bu durumda: \[ 2a = 2 \times 6.999 = 13.998 \] Şimdi \( b \) değerini minimize etmeliyiz. \( b \) için en küçük değer olan -1'e yaklaşmalıyız. Yani \( b = -0.999 \) alalım. Bu durumda: \[ -b = -(-0.999) = 0.999 \] Şimdi \( 2a - b \) ifadesini hesaplayalım: \[ 2a - b = 13.998 + 0.999 = 14.997 \] Bu değeri tam sayıya yuvarladığımızda, en büyük tam sayı değeri 14 olacaktır. Sonuç olarak, \( 2a - b \) ifadesinin alabileceği en büyük tam sayı değeri: \[ \boxed{14} \]

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Bonus Knowledge

Verilen eşitsizlikleri kullanarak, ifademizi maksimuma çıkarmak için \(2a - b\) ifadesini inceleyelim. Öncelikle, \(a\) en büyük değeri 7’ye çok yaklaşabileceğinden, \(a\) için mümkün olan en yüksek değeri alarak, \(a = 7\) (ama tam olarak 7 değil) değerini düşünelim. \(2a\) ifadesi bu durumda yaklaşık olarak \(2 \times 7 = 14\) olur. Şimdi ise \(b\) ifadesinin en küçük değerini almak istiyoruz. En küçük değeri olan \(-1\) için, \(-b\) ifadesi \(-(-1)\) yani \(1\) olur. Dolayısıyla, \(2a - b\) ifadesini maksimum \(14 + 1 = 15\) buluyoruz. Ama en büyük tam sayı değeri 15'ten bir önceki değerdir, bu da 14’tür. Yani \(2a - b\) ifadesinin alabileceği en büyük tam sayı değeri 14'tür.

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