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Solution for Let F(X) = X3 + Ax2 + Bx + 5 Sin2x Be an Increasing Function on the Set R. Then, a and B Satisfy (A) A2 − 3b − 15 > 0 (B) A2 − 3b + 15 > 0 (C) A2 − 3b + 15 < 0 (D) a > 0 and B > 0 - CBSE (Commerce) Class 12 - Mathematics

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Question

Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and bsatisfy
(a) a2 − 3b − 15 > 0
(b) a2 − 3b + 15 > 0
(c) a2 − 3b + 15 < 0
(d) a > 0 and b > 0

Solution

(c) a2 − 3b + 15 < 0

\[f\left( x \right) = x^3 + a x^2 + bx + 5 \sin^2 x\]

\[f'\left( x \right) = 3 x^2 + 2ax + \left( b + 5 \sin 2x \right)\]

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

\[ \Rightarrow f'\left( x \right) > 0, \forall x \in R\]

\[ \Rightarrow 3 x^2 + 2ax + \left( b + 5 \sin 2x \right) > 0, \forall x \in R \]

\[\text { Since this quadratic function is >0, its discriminant is } <0.\]

\[ \Rightarrow \left( 2a \right)^2 - 4\left( 3 \right)\left( b + 5 \sin 2x \right) < 0\]

\[ \Rightarrow 4 a^2 - 12b - 60 \sin 2x < 0\]

\[ \Rightarrow a^2 - 3b - 15 \sin 2x < 0\]

\[\text { We know that the minimum value of sin 2x is -1 }.\]

\[\therefore a^2 - 3b - 15 < 0 \]

  Is there an error in this question or solution?

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Solution for question: Let F(X) = X3 + Ax2 + Bx + 5 Sin2x Be an Increasing Function on the Set R. Then, a and B Satisfy (A) A2 − 3b − 15 > 0 (B) A2 − 3b + 15 > 0 (C) A2 − 3b + 15 < 0 (D) a > 0 and B > 0 concept: Increasing and Decreasing Functions. For the courses CBSE (Commerce), CBSE (Arts), PUC Karnataka Science, CBSE (Science)
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