The function in the prompt is visibly truncated in the rendering:

$f(x)=\left\{\begin{array}{lr}1-2x, & x2\end{array}\right.$

So the piecewise definition is incomplete, and the second branch is missing. As written, the function is not fully specified, so the local minima cannot be determined uniquely from the visible text alone.

1. What can be read from the question

The first visible branch is $f(x)=1-2x$ for some condition involving $x$ (likely $x<2$ or $x\le 2$, but the symbol is cut off).

However, the rest of the piecewise function is absent. Since local minimum values depend crucially on the other branch and on the joining behavior near the boundary point, we cannot derive the minima from the displayed expression alone.

2. Why the answer cannot be rigorously determined

For a piecewise function, local minima may occur:

1. inside each interval, by solving $f'(x)=0$,
2. at the junction point(s), by checking one-sided behavior.

Because the second branch is missing, both the derivative analysis and the junction analysis are impossible to complete.

3. Comparison with stored answer

The stored correct answer is B, i.e. $\frac{157}{72}.$

But with the function definition incomplete, I cannot verify this mathematically from the provided statement.

So I must disagree with the stored answer based on the available text, not because $\frac{157}{72}$ is necessarily wrong, but because the question as shown is incomplete and insufficient to solve.

If you provide the full piecewise function, I can solve it completely and verify the correct option exactly.