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Question
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
Options
k < 3
k ≤ 3
k > 3
k ≥ 3
Solution
k > 3
\[f\left( x \right) = k x^3 - 9 x^2 + 9x + 3\]
\[f'\left( x \right) = 3k x^2 - 18x + 9\]
\[ = 3 \left( k x^2 - 6x + 3 \right)\]
\[\text { Given:f(x) is monotonically increasing in every interval }.\]
\[ \Rightarrow f'\left( x \right) > 0\]
\[ \Rightarrow 3 \left( k x^2 - 6x + 3 \right) > 0\]
\[ \Rightarrow \left( k x^2 - 6x + 3 \right) > 0\]
\[ \Rightarrow k > 0 \text { and } \left( - 6 \right)^2 - 4\left( k \right)\left( 3 \right) < 0 \left[ \because a x^2 + bx + c > 0 \Rightarrow a > 0 \text { and Disc} < 0 \right]\]
\[ \Rightarrow k > 0 \text { and } \left( - 6 \right)^2 - 4\left( k \right)\left( 3 \right) < 0\]
\[ \Rightarrow k > 0 \text { and }36 - 12k < 0\]
\[ \Rightarrow k > 0 \text { and }12k > 36\]
\[ \Rightarrow k > 0 \text { and } k > 3\]
\[ \Rightarrow k > 3\]
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