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Solution for If the Function F(X) = Kx3 − 9x2 + 9x + 3 is Monotonically Increasing in Every Interval, Then (A) K < 3 (B) K ≤ 3 (C) K > 3 (D) K ≥ 3 - CBSE (Science) Class 12 - Mathematics

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Question

If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
(a) k < 3
(b) k ≤ 3
(c) k > 3
(d) k ≥ 3

Solution

(c) k > 3

\[f\left( x \right) = k x^3 - 9 x^2 + 9x + 3\]

\[f'\left( x \right) = 3k x^2 - 18x + 9\]

\[ = 3 \left( k x^2 - 6x + 3 \right)\]

\[\text { Given:f(x) is monotonically increasing in every interval }.\]

\[ \Rightarrow f'\left( x \right) > 0\]

\[ \Rightarrow 3 \left( k x^2 - 6x + 3 \right) > 0\]

\[ \Rightarrow \left( k x^2 - 6x + 3 \right) > 0\]

\[ \Rightarrow k > 0 \text { and } \left( - 6 \right)^2 - 4\left( k \right)\left( 3 \right) < 0 \left[ \because a x^2 + bx + c > 0 \Rightarrow a > 0 \text { and Disc} < 0 \right]\]

\[ \Rightarrow k > 0 \text { and } \left( - 6 \right)^2 - 4\left( k \right)\left( 3 \right) < 0\]

\[ \Rightarrow k > 0 \text { and }36 - 12k < 0\]

\[ \Rightarrow k > 0 \text { and  }12k > 36\]

\[ \Rightarrow k > 0 \text { and } k > 3\]

\[ \Rightarrow k > 3\]

  Is there an error in this question or solution?

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Solution for question: If the Function F(X) = Kx3 − 9x2 + 9x + 3 is Monotonically Increasing in Every Interval, Then (A) K < 3 (B) K ≤ 3 (C) K > 3 (D) K ≥ 3 concept: Increasing and Decreasing Functions. For the courses CBSE (Science), PUC Karnataka Science, CBSE (Arts), CBSE (Commerce)
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