The sum of the maximum and minimum values of the function f(x) =...

Given,

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

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

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

\left(\text{as}\left[x - 1\right]\text{=}\left[x\right]\text{-1 where}\left[.\right]\text{is greatest integer function}\right)$</p><p>Now critical points of</p><p>$f\left(x\right) = \dfrac{7}{5}, \sqrt{5} - 1, \sqrt{6} - 1, \sqrt{7} - 1, \sqrt{8} - 1, 2$</p><p>$∴$Maximum or minimum value of$f\left(x\right)$occur at critical points or boundary points.</p><p>So,$f\left(\dfrac{5}{4}\right) = \dfrac{3}{4} + 4 = \dfrac{19}{4}$and$f\left(\dfrac{7}{5}\right) = 0 + 4 = 4$</p><p>As both \left|5x - 7\right|$ and ${x}^{2} + 2x$ are increasing in nature after $x = \dfrac{7}{5}$

So, $f\left(2\right) = \left|10 - 7\right| + \left[4 + 4\right] = 3 + 8 = 11$

∴ f{\left(\dfrac{7}{5}\right)}_\min = 4 and f{\left(2\right)}_\max = 11

So sum of minimum and maximum value is $4 + 11 = 15$