1. **Problem Statement:** Given that the curve $y = f(|L|)$ intersects the coordinate axes exactly twice, determine which of the statements I, II, and III are necessarily true. 2. **Understanding the problem:** - The curve $y = f(|L|)$ means the function $f$ is applied to the absolute value of $L$. - Intersecting the coordinate axes means points where the curve crosses the $x$-axis ($y=0$) or the $y$-axis ($x=0$). - The curve $y = f(|L|)$ intersects the axes exactly twice. 3. **Analyze each statement:** **I. The curve $y = |f(L)|$ intersects the coordinate axes exactly once.** - $y = |f(L)|$ means the output is always non-negative. - The $x$-axis intersections occur where $y=0$, so where $f(L)=0$. - The $y$-axis intersection is at $L=0$, so $y=|f(0)|$. - Since $y$ is non-negative, the curve can only touch or cross the $x$-axis where $f(L)=0$. - The number of intersections depends on zeros of $f(L)$. - No guarantee it is exactly one. **II. The curve $y = f(|L|)$ intersects coordinate axes exactly once.** - Given in the problem that $y = f(|L|)$ intersects axes exactly twice. - So this statement says exactly once, which contradicts the problem. - Therefore, II is false. **III. The curve $|y| = f(|L|)$ intersects the coordinate axes exactly twice.** - $|y| = f(|L|)$ means $y = \\pm f(|L|)$. - Intersections with axes are where $y=0$ or $L=0$. - Since $f(|L|)$ is the same as in the original problem, and absolute value on $y$ allows both positive and negative $y$ values. - The number of intersections remains the same as for $y = f(|L|)$. - So III is true. 4. **Conclusion:** - Only statement III is necessarily true. 5. **Answer choice:** - D. III