The sum of all local minimum values of the function \( f(x) \) as defined below is:
\[
f(x) = \left\{
\begin{array}{ll}
1 - 2x & \text{if } x < -1 \\
\frac{1}{3}(7 + 2|x|) & \text{if } -1 \leq x \leq 2 \\
\frac{11}{18} (x-4)(x-5) & \text{if } x > 2
\end{array}
\right.
\]
\[ f(x) = \left\{ \begin{array}{ll} 1 - 2x & \text{if } x < -1 \\ \frac{1}{3}(7 + 2|x|) & \text{if } -1 \leq x \leq 2 \\ \frac{11}{18} (x-4)(x-5) & \text{if } x > 2 \end{array} \right. \]
Show Hint
- \(\frac{167}{72}\)
- \(\frac{171}{72}\)
- \(\frac{131}{72}\)
- \(\frac{157}{72}\)
The Correct Option is A
Solution and Explanation
Identify critical points within the domain of each piecewise segment.
Step 2: Calculate the minimum values.
Calculate the values at critical points and sum them up.
Conclusion:
The sum of all local minimum values is \(\frac{167}{72}\).
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