$$f(x) = |5x - 7| + [{x^2} + 2x]$$

$$ = |5x - 7| + [{(x + 1)^2}] - 1$$

Critical points of

$$f(x) = {7 \over 5},\sqrt 5 - 1,\,\sqrt 6 - 1,\,\sqrt 7 - 1,\,\sqrt 8 - 1,\,2$$

$$\therefore$$ Maximum or minimum value of $$f(x)$$ occur at critical points or boundary points

$$\therefore$$ $$f\left( {{5 \over 4}} \right) = {3 \over 4} + 4 = {{19} \over 4}$$

$$f\left( {{7 \over 5}} \right) = 0 + 4 = 4$$

as both $$|5x - 7|$$ and $${x^2} + 2x$$ are increasing in nature after $$x = {7 \over 5}$$

$$\therefore$$ $$f(2) = 3 + 8 = 11$$

$$\therefore$$ $$f{\left( {{7 \over 5}} \right)_{\min }} = 4$$ and $$f{(2)_{\max }} = 11$$

Sum is $$4 + 11 = 15$$