Question

Let $F(x)=x^3+a{x}^2+bx+5si{n}^{2}x$ be an increasing function in the set of real number $R$. 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. $a>0$ and $b>0$

Answer 1

Answer: (C)

We have $f(x)=x^3+a{x}^2+bx+5si{n}^{2}x$
$\Rightarrow f^{\prime }(x)3x^2+2ax+b+5sin2x$
$\because f(x)$ is an increasing function
$\therefore f^{\prime }\left(x\right)>0\Rightarrow 3x^2+2ax+b+5sin2x>0,$
$\left(\because sin2x<1\right)$
$\therefore 0<3x^2+2ax+b+5sin2x<3x^2+2ax+b+5$
$\Rightarrow 3x^2+2ax+b+5>0$
$\Rightarrow 4a^2+4.3\left(b+5\right)<0\Rightarrow a^2+3b-15<0$
[$\because a{x}^2+bx+c>0$ or all real $x$ if . $a>0$ and discriminant $<0$]