The sum of the maximum and minimum values of the function $f(x)=|5 x-7|+\left[x^{2}+2 x\right]$ in the interval $\left[\frac{5}{4}, 2\right]$, where $[t]$ is the greatest integer $\leq t$, is ______________.

<p>$f(x) = |5x - 7| + [{x^2} + 2x]$</p> <p>$= |5x - 7| + [{(x + 1)^2}] - 1$</p> <p>Critical points of</p> <p>$f(x) = {7 \over 5},\sqrt 5 - 1,\,\sqrt 6 - 1,\,\sqrt 7 - 1,\,\sqrt 8 - 1,\,2$</p> <p>$\therefore$ Maximum or minimum value of $f(x)$ occur at critical points or boundary points</p> <p>$\therefore$ $f\left( {{5 \over 4}} \right) = {3 \over 4} + 4 = {{19} \over 4}$</p> <p>$f\left( {{7 \over 5}} \right) = 0 + 4 = 4$</p> <p>as both $|5x - 7|$ and ${x^2} + 2x$ are increasing in nature after $x = {7 \over 5}$</p> <p>$\therefore$ $f(2) = 3 + 8 = 11$</p> <p>$\therefore$ $f{\left( {{7 \over 5}} \right)_{\min }} = 4$ and $f{(2)_{\max }} = 11$</p> <p>Sum is $4 + 11 = 15$</p>