If $f(x)=x^3+ax^2+bx+5\sin^2x$ is a strictly increasing function on the set of real numbers then $a$ and $b$ must satisfy the relation If $f(x)=x^3+ax^2+bx+5\sin^2x$ is a strictly increasing function on the set of real numbers then $a$ and $b$ must satisfy the relation
$(A)a^2-3b+15\leq0$
$(B)a^2-3b+20\leq0$
$(C)a^2-3b+25\leq0$
$(D)a^2-3b+30\leq0$

Since $f(x)=x^3+ax^2+bx+5\sin^2x$ is a strictly increasing function.
So $f'(x)=3x^2+2ax+b+5\sin2x>0$
I am stuck here.

Since 
\begin{align*}
3x^2+2ax+b+5\sin2x \geq 3x^2+2ax+b-5.
\end{align*}
Consider the quadratic expression $3x^2+2ax+b-5$. For this to be strictly positive, the discriminant should be $\leq 0$. Thus
\begin{align*}
4a^2-12(b-5) & \leq 0\\
a^2-3b+15 & \leq 0.
\end{align*}
