Question

If $f\left(x\right)=x^3+a{x}^2+bx+5{\sin }^{2}x$ be a decreasing function in R. Then a and b satisfy the condition

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

Answer 1

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

$f\left(x\right)=x^3+a{x}^2+bx+5{\sin }^{2}x$
$f^{{}^{\prime }}\left(x\right)=3x^2+2ax+b+5\sin 2x<0asf\left(x\right)$ is $\downarrow$
But $\sin 2x\ge -1$
$\therefore 3x^2+2ax+b-5\le 3x^2+2ax+b+5\sin 2x<0$
$\Rightarrow 3x^2+2ax+(b-5)<0$
$\Rightarrow 4a^2-4\ast 3(b-5)<0$
$\Rightarrow a^2-3b+15<0$