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Example 31 - Find local minimum value of f(x) = 3 + |x| - Local maxima and minima

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  1. Chapter 6 Class 12 Application of Derivatives
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Example 31 (Method 1) Find local minimum value of the function f given by f (๐‘ฅ) = 3 + |๐‘ฅ|, ๐‘ฅ โˆˆ R. f (๐‘ฅ) = 3 + |๐‘ฅ| ๐‘ฅ โˆˆ R Since Value of ๏ท๐‘ฅ๏ทฏโ‰ฅ0 So, Minimum value of ๏ท๐‘ฅ๏ทฏ=0 โ‡’ Minimum value of f (๐‘ฅ) = 3 + Minimum value of ๏ท๐‘ฅ๏ทฏ = 3 + 0 = 3 Hence minimum value of f (๐‘ฅ) = 3 We cannot find any maximum value. Example 31 (Method 2) Find local minimum value of the function f given by f (๐‘ฅ) = 3 + |๐‘ฅ|, ๐‘ฅ โˆˆ R. f (๐‘ฅ) = 3 + |๐‘ฅ| We know that ๏ท๐‘ฅ๏ทฏ=๏ท๏ทโˆ’&๐‘ฅ, ๐‘ฅ<0๏ทฎ&๐‘ฅ, ๐‘ฅโ‰ฅ0๏ทฏ๏ทฏ Hence, f (๐‘ฅ) =๏ท๏ท3โˆ’&๐‘ฅ, ๐‘ฅ<0๏ทฎ3+๐‘ฅ, ๐‘ฅโ‰ฅ0๏ทฏ๏ทฏ ๐‘ฅ = 0 is only critical point We check sign of fโ€™ (๐‘ฅ) at ๐‘ฅ = 0 Since sign of fโ€™(x) changes from negative to positive, x = 0 is point of Maxima & f (๐‘ฅ) is Minimum at ๐‘ฅ = 0 Minimum value of f (๐‘ฅ) f (0) = 3 + |0|= 3 Thereโ€™s no Maximum value of f (๐’™) Example 31 (Method 3) Find local minimum value of the function f given by f (๐‘ฅ) = 3 + |๐‘ฅ|, ๐‘ฅ โˆˆ R. f (๐‘ฅ) = 3 + |๐‘ฅ| From graph, f (๐‘ฅ) is Minimum at ๐‘ฅ = 0 So, Minimum value of f (๐‘ฅ) = f (0) = 3

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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