CBSE (Science) Class 12CBSE
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

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

Login
Create free account


      Forgot password?

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)

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

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

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

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

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

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

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

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

\[ \Rightarrow 36 k^2 - 36 < 0\]

\[ \Rightarrow k^2 - 1 < 0\]

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

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

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

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

\[ \Rightarrow k > - 1 \text { and } k < 1\]

\[ \Rightarrow - 1 < k < 1\]

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

  Is there an error in this question or solution?

APPEARS IN

Solution for 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 concept: Increasing and Decreasing Functions. For the courses CBSE (Science), PUC Karnataka Science, CBSE (Arts), CBSE (Commerce)
S
View in app×