Question

The sum of all local minimum values of the function \( f(x) \) as defined below is:

\[ f(x) = \left\{ \begin{array}{ll} 1 - 2x & \text{if } x < -1 \\ \frac{1}{3}(7 + 2|x|) & \text{if } -1 \leq x \leq 2 \\ \frac{11}{18} (x-4)(x-5) & \text{if } x > 2 \end{array} \right. \]

• A. \(\frac{167}{72}\)
• B. \(\frac{171}{72}\)
• C. \(\frac{131}{72}\)
• D. \(\frac{157}{72}\)

Hint:

Piecewise functions may have multiple local minima or maxima, examine all segments thoroughly.

Answer 1

The Correct Option is A

Step 1: Analyze each piece.
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 \(\frac{167}{72}\).