Piecewise function: continuous, differentiable and tangent line

Problem statement

Let the function be

$$f(x)=\begin{cases}3x^2-\sin(ax)+b & \text{if } x<0\\[4pt] \dfrac{2e^x}{1+x^2} & \text{if } x\ge 0\end{cases}$$

a) (1 mark) Find the values of $a$ and $b$ so that the function is continuous and differentiable at $x = 0$.

b) (1 mark) Find the equation of the tangent line to the graph of the function at the point with abscissa $x = 2$.

Source: Madrid university entrance exam (PAU, Spain) · 2025-2026 · Ordinary session · Mathematics · Question 5.2. Official exam set by the public universities of the Community of Madrid, to whom it belongs; translated from Spanish and reproduced with its source in order to explain it for teaching purposes.