Question

Let $f\left(x\right)=x^3+a{x}^2+bx+5{sin}^{2}x$ be an increasing function on the set R. Then, a and b satisfy

• A. $a^2-3b-15>0$
• B. $a^2-3b+15>0$
• C. $a^2-3b+15<0$
• D. $a>0$ and$b>0$

Answer 1

The correct option is C $a^2-3b+15<0$

$f\left(x\right)=x^3+a{x}^2+bx+5si{n}^{2}x$ $x\in [0,\frac{\pi }{2}]$

Given: Function$f\left(x\right)$ is increasing on the set $R$

Therefore,

$f^{{}^{\prime }}\left(x\right)>0$

$3x^2+2ax+b-5sin2x>0$

Since,

$f^{{}^{\prime }}\left(x\right)>0;D<0$ , Here $D$ is discriminant

$(2a{)}^2-(4\times 3)\times (b-5sin2x)<0$

$4a^2-12b+60sin2x<0$

$a^2-3b+15sin2x<0$

Since, $x\in [0,\frac{\pi }{2}]$

Therefore, $sin2x>0$

$sin2x$ is always positive

Hence,

$a^2-3b+15<0$