Question

If $f\left(x\right)=x^3+a{x}^2+bx+5{\sin }^{2}x$ is a strictly increasing function on the set of real numbers, then $a$ and $b$ must satisfy the relation

• A. $a^2-3b+15\le 0$
• B. $a^2-3b+20\le 0$
• C. $a^2-3b+25\le 0$
• D. $a^2-3b+30\le 0$

Answer 1

The correct option is A $a^2-3b+15\le 0$

We have $f^{\prime }\left(x\right)=3x^2+2ax+b+5\sin 2x\ge 0$
As $-1\le \sin 2x\le 1,$
Hence $f^{\prime }\left(x\right)=3x^2+2ax+b-5\ge 0\forall x\in R$
So, $D\le 0$
$\Rightarrow 4a^2-12(b-5)\le 0$
$\Rightarrow a^2-3b+15\le 0$