Find all $a,b \in \mathbb{R}$ such that $f: \mathbb{R} \to \mathbb{R}, \ f(x) = \sin(ax) + e^{bx^2} $is convex

Question

What I tried so far:

• using the definition of convexity → very general, not really good to use?
• calculate $f''(x)$ and find $a, b$ such that $ f''(x) \geq 0 \ \forall x$ → difficult to find all $a,b$ such that it is always positive.

\begin{align} f'(x) &= a \cdot \cos(ax) + 2xb \cdot e^{bx^2} \\ f''(x) &= -a^2 \cdot \sin(ax) + 4 x^2 b^2 \cdot e^{bx^2} + 2b \cdot e^{bx^2} \\ &= -a^2 \cdot \sin(ax) + e^{bx^2}\cdot (4 x^2b^2 + 2b) \end{align}

Is there anything else that I could / should try when proving convexity?