Question

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

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

Answer 1

Answer: (A)

$a^2-3b-15>0$

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

Given that $f\left(x\right)$ is incresing on $R$

Therefore, $\to f^{\prime }\left(x\right)\ge 0$

$\Rightarrow f^{\prime }\left(x\right)=3x^2+2ax+b+10\sin x\cos x$

$\Rightarrow f^{\prime }\left(x\right)=3x^2+2ax+b+5\sin 2x\ge 0$

For roots to exist, $\Delta \ge 0$

Thereore, $\Rightarrow (2a{)}^2-4(3\left)\right(b+5\sin 2x)\ge 0$

$\Rightarrow 4a^2-4\left(3\right)(b+5\sin 2x)\ge 0$

$\Rightarrow a^2-3b-15\sin 2x\ge 0$

Maximum value of $\sin 2x=1$

Therefore,$\Rightarrow a^2-3b-15\ge 0$