Suppose $$𝑦 = 𝑦𝑥$$ be the solution curve to the differential equation $$\frac{dy}{dx}-y=2-e^{-x}$$ such that $$\lim_{x \rightarrow \infty} yx$$ If $$𝑎$$ and $$𝑏$$ are respectively the $$𝑥 -$$ and $$𝑦 -$$ intercept of the tangent to the curve at $$𝑥 = 0$$, then the value of $$𝑎 - 4𝑏$$ is equal to _______.

We need to evaluate $$\displaystyle\int_0^6 f(x)\,dx$$ where $$f(x) = \max\{|x+1|, |x+2|, \ldots, |x+5|\}$$.

For any $$x \in [0, 6]$$ and $$k = 1, 2, 3, 4, 5$$: $$x + k > 0$$, so $$|x + k| = x + k$$.

Among $$x + 1, x + 2, x + 3, x + 4, x + 5$$, the largest is $$x + 5$$.

Therefore $$f(x) = x + 5$$ for all $$x \in [0, 6]$$.

$$\int_0^6 (x + 5)\,dx = \left[\dfrac{x^2}{2} + 5x\right]_0^6 = \left(\dfrac{36}{2} + 30\right) - 0 = 18 + 30 = 48$$

The answer is $$48$$.