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

Answer 1

(c) a² − 3b + 15 < 0


$$\begin{gathered} f\left(x\right)=x^3+a{x}^2+bx+5{\sin }^{2}x \\ f\text{'}\left(x\right)=3x^2+2ax+\left(b+5\sin 2x\right) \\ \text{Given:}f\left(x\right)\text{is increasing on}\mathit{\text{R}}\text{.} \\ \Rightarrow f\text{'}\left(x\right)>0,\forall x\in R \\ \Rightarrow 3x^2+2ax+\left(b+5\sin 2x\right)>0,\forall x\in R \\ \text{Since this quadratic function is >0, its discriminant is <0.} \\ \Rightarrow {\left(2a\right)}^2-4\left(3\right)\left(b+5\sin 2x\right)<0 \\ \Rightarrow 4a^2-12b-60\sin 2x<0 \\ \Rightarrow a^2-3b-15\sin 2x<0 \\ \text{We know that the minimum value of sin 2x is -1.} \\ \text{∴}{a}^2-3b-15<0 \end{gathered}$$