Question

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

• 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

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

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

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

Since $f\left(x\right)$ is increasing.

Therefore,

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

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

Minimum value of $\sin 2x=-1$

Therefore,

$3x^2+2ax+b-5>0$

Since the above quadratic equation is $>0$, therefore, discriminant should be $<0$.

Therefore,

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

$4a^2-12b+60<0$

$a^2-3b+15<0$

Hence for $f\left(x\right)$ to be increasing function, $a$ and $b$ satisfy the equation $a^2-3b+15<0$.