The sum of all local minimum values of the function $$ f(x)=\left\{\begin{array}{lr} 1-2 x, & x \lt -1 \\ \frac{1}{3}(7+2|x|), & -1 \leq x \leq 2 \\ \frac{11}{18}(x-4)(x-5), & x\gt2 \end{array}\right. $$ is

• Correct answer
  A

  $\frac{157}{72}$
• B

  $\frac{131}{72}$
• C

  $\frac{171}{72}$
• D

  $\frac{167}{72}$

Solution

$f(x)=\left\{\begin{array}{cl}1-2 x, & x < -1 \\ \frac{1}{3}(7-2 x), & -1 \leq x \leq 2 \\ \frac{1}{3}(7+2 x) & 0 \leq x < 2 \\ \frac{11}{18}(x-4)(x-5), & x>2\end{array}\right.$
$\therefore$ Local minimum values at $\mathrm{A} \& \mathrm{~B}$ $\begin{aligned} & \frac{7}{3}-\frac{11}{72} \\ & \Rightarrow \frac{168-11}{72} \Rightarrow \frac{157}{72} \end{aligned}$

Hence, the correct answer is option: 'A'.