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# Solution for If the Function F(X) = X3 − 9kx2 + 27x + 30 is Increasing on R, Then (A) −1 ≤ K < 1 (B) K < −1 Or K > 1 (C) 0 < K < 1 (D) −1 < K < 0 - CBSE (Science) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

#### Question

If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
(a) −1 ≤ k < 1
(b) k < −1 or k > 1
(c) 0 < k < 1
(d) −1 < k < 0

#### Solution

(a)

$f\left( x \right) = x^3 - 9k x^2 + 27x + 30$

$f'\left( x \right) = 3 x^2 - 18kx + 27$

$= 3 \left( x^2 - 6kx + 9 \right)$

$\text { Given: f(x) is increasing on R } .$

$\Rightarrow f'\left( x \right) > 0 \text { for all } x \in R$

$\Rightarrow 3 \left( x^2 - 6kx + 9 \right) > 0 \text { for all } x \in R$

$\Rightarrow x^2 - 6kx + 9 > 0 \text { for all } x \in R$

$\Rightarrow \left( - 6k \right)^2 - 4\left( 1 \right)\left( 9 \right) < 0 \left[ \because a x^2 + bx + c > 0 for all x \in R \Rightarrow a > 0 and Disc < 0 \right]$

$\Rightarrow 36 k^2 - 36 < 0$

$\Rightarrow k^2 - 1 < 0$

$\Rightarrow \left( k + 1 \right)\left( k - 1 \right) < 0$

$\text { It can be possible when } \left( k + 1 \right) < 0 \text { and } \left( k - 1 \right) > 0 .$

$\Rightarrow k < - 1 \text { and } k > 1 (\text { Not possible })$

$or \left( k + 1 \right) > 0 \text { and } \left( k - 1 \right) < 0$

$\Rightarrow k > - 1 \text { and } k < 1$

$\Rightarrow - 1 < k < 1$

$\text { Disclaimer: (a) part should be } - 1 < k < 1 \text { instead of }-1 \leq k < 1 .$

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Solution If the Function F(X) = X3 − 9kx2 + 27x + 30 is Increasing on R, Then (A) −1 ≤ K < 1 (B) K < −1 Or K > 1 (C) 0 < K < 1 (D) −1 < K < 0 Concept: Increasing and Decreasing Functions.
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