Question

Let $f\left(x\right)=x^3+a{x}^2+bx+5{\sin }^{2}x$ be an increasing fuction in the set of real numbers $R$. Then a & b satisfy the condition:

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

Answer 1

Answer: (B)

$a^2-3b+15\le 0$

$f\left(x\right)=x^3+a{x}^2+bx+5{\sin }^{2}x$

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

$f^{\prime }\left(x\right)=3x^2+a\left(2x\right)+b+5\left(2\sin x\right)\left(\cos x\right)$

$=3x^2+2ax+b+10\sin x\cos x>0$

$=3x^2+2ax+b+5\left(2\sin x\cos x\right)>0$

$3x^2+2ax+b+5\sin 2x>0$

$3x^2+2ax+\left(b-5\right)>0$

$a>0$, $D\le 0$

$D\le 0$

$b^2-4ac\le 0$

${\left(2a\right)}^2-4\times 3\times \left(b-5\right)\le 0$

$4a^2-12\left(b-5\right)\le 0$

$4\left(a^2-3b+15\right)\le 0$

$a^2-3b+15\le 0$