Example 8 - Show f(x) = x3 - 3x2 + 4x is strictly increasing - Examples

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  1. Chapter 6 Class 12 Application of Derivatives
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Example 8 Show that the function f given by f (๐‘ฅ) = ๐‘ฅ3 โ€“ 3๐‘ฅ2 + 4๐‘ฅ, ๐‘ฅ โˆˆ R is strictly increasing on R. f(๐‘ฅ) = ๐‘ฅ3 โ€“ 3๐‘ฅ2 + 4๐‘ฅ fโ€™ (๐‘ฅ) = 3๐‘ฅ2 โ€“ 3.2๐‘ฅ + 4 fโ€™ (๐‘ฅ) = 3x2 โ€“ 6๐‘ฅ + 4 fโ€™ (๐‘ฅ) = 3x2 โ€“ 6๐‘ฅ + 3 + 1 fโ€™ (๐‘ฅ) = 3 (๐‘ฅ2 โ€“ 2๐‘ฅ + 1) + 1 fโ€™ (๐‘ฅ) = 3 (๐‘ฅ โ€“ 1)2 + 1 As square is a positive number, The value of fโ€™(๐‘ฅ) will be always positive for every real number Hence fโ€™(๐‘ฅ) > 0 for all ๐‘ฅ โˆˆ R โˆด f(๐’™) is strictly increasing

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