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

$\begin{array}{l}\frac{dy}{dx}=3x^2+2ax+b+5\sin 2x=\text{positive}\mathrm{\forall }x\in R\\ =3x^2+2ax+b-5\end{array}$

as the least value of $\sin 2x=-1$

The expression (1) will be of the same sign as of the  first term i.e. positive provided its  $\mathrm{\Delta }=\mathrm{negative}$

$\begin{array}{r}4a^2-4.3(b-5)<0\\ a^2-3b+15<0\Rightarrow \left(\mathrm{c}\right)\end{array}$