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

$x^2+2x$  increases in $[\frac{5}{4},2]$

So,  $[x^2+2x]$ also increases
$|5x-7|$ has minima at  $x=\frac{7}{5}$
where $[x^2+2x]$  is constant
so,  $f_{\min }=f\left(\frac{7}{5}\right)=0+4=4$
and  $f_{\max }=f\left(2\right)=3+8=11$
Required sum = 15