Mathematics - Functions Question with Solution | TestHub

Given,

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

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

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

$\left(\text{as }\left[x-1\right]\text{=}\left[x\right]\text{-1 where }\left[.\right]\text{ is greatest integer function}\right)$

Now critical points of

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

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

So, $f\left(\frac{5}{4}\right)=\frac{3}{4}+4=\frac{19}{4}$ and $f\left(\frac{7}{5}\right)=0+4=4$

As both $\left|5x-7\right|$ and $x^2+2x$ are increasing in nature after $x=\frac{7}{5}$

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

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

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