If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then

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f\left( x ight) = x^3 - 9k x^2 + 27x + 30 f'\left( x ight) = 3 x^2 - 18kx + 27 = 3 \left( x^2 - 6kx + 9 ight) ext { Given: f(x) is increasing on R } . \Rightarrow f'\left( x ight) > 0 ext { for all } x \in R \Rightarrow 3 \left( x^2 - 6kx + 9 ight) > 0 ext { for all } x \in R \Rightarrow x^2 - 6kx + 9 > 0 ext { for all } x \in R \Rightarrow \left( - 6k ight)^2 - 4\left( 1 ight)\left( 9 ight) < 0 \left[ \because a x^2 + bx + c > ext { 0 for all }x \in R \Rightarrow a > ext{0 and Disc}< 0 ight] \Rightarrow 36 k^2 - 36 < 0 \Rightarrow k^2 - 1 < 0 \Rightarrow \left( k + 1 ight)\left( k - 1 ight) < 0 ext { It can be possible when } \left( k + 1 ight) < 0 ext { and } \left( k - 1 ight) > 0 . \Rightarrow k < - 1 ext { and } k > 1 ( ext { Not possible }) or \left( k + 1 ight) > 0 ext { and } \left( k - 1 ight) < 0 \Rightarrow k > - 1 ext { and } k < 1 \Rightarrow - 1 < k < 1 ext { Disclaimer: (a) part should be } - 1 < k < 1 ext { instead of }-1 \leq k < 1 .

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