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Question
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Options
a2 − 3b − 15 > 0
a2 − 3b + 15 > 0
a2 − 3b + 15 < 0
a > 0 and b > 0
Solution
a2 − 3b + 15 < 0
Explanation:
\[f\left( x \right) = x^3 + a x^2 + bx + 5 \sin^2 x\]
\[f'\left( x \right) = 3 x^2 + 2ax + \left( b + 5 \sin 2x \right)\]
\[\text {Given}:f\left( x \right)\text { is increasing on R }.\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in R\]
\[ \Rightarrow 3 x^2 + 2ax + \left( b + 5 \sin 2x \right) > 0, \forall x \in R \]
\[\text { Since this quadratic function is >0, its discriminant is } <0.\]
\[ \Rightarrow \left( 2a \right)^2 - 4\left( 3 \right)\left( b + 5 \sin 2x \right) < 0\]
\[ \Rightarrow 4 a^2 - 12b - 60 \sin 2x < 0\]
\[ \Rightarrow a^2 - 3b - 15 \sin 2x < 0\]
\[\text { We know that the minimum value of sin 2x is−1}.\]
\[\therefore a^2 - 3b + 15 < 0 \]
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