Question

For positive real numbers 𝑆 and 𝐾, the function 𝐻𝐾(𝑆) is defined as:
𝐻𝐾(𝑆) = max(𝑆−𝐾, 0). The max function is defined as:
max(𝑎, 𝑏) = {𝑎,
when 𝑎> 𝑏
𝑏,
when 𝑎≤𝑏
The graph below shows the plot of a function 𝑁(𝑆) versus 𝑆.
𝑁(𝑆) can be expressed as _____.

• A. \(H_{10}(S)-H_{20}(S)\)
• B. \(H_{10}(S)-2H_{20}(S)\)
• C. \(-H_{10}(S)+H_{20}(S)\)
• D. \(H_{15}(S)-H_{20}(S)\)

Hint:

Each \(H_K(S)\) is a ramp that switches on at \(S=K\); pick a combination whose slopes cancel exactly beyond \(S=20\) so the result flattens at height 10, and check with a test point like \(S=25\) to eliminate the wrong options.

Answer 1

The Correct Option is A

Think of \(H_K(S) = \max(S-K, 0)\) as a hinge that stays flat at zero until \(S=K\), then bends upward with slope 1 forever after. Combining two such hinges with opposite signs lets you build a shape that turns on, rises, and then flattens out again - exactly the graph shown for \(N(S)\): flat at 0 up to \(S=10\), a straight climb up to a height of 10 at \(S=20\), then flat at 10 forever after.

Building the combination: To get a ramp that starts rising at \(S=10\), you need a \(+H_{10}(S)\) term (this alone would rise forever). To cancel that rise beyond \(S=20\) and freeze the value, you subtract a second hinge that switches on at \(S=20\) with the same slope: \(-H_{20}(S)\). Beyond \(S=20\), both hinges rise at slope 1, so their difference stops changing - it locks in at whatever value was reached at \(S=20\), which is \(20-10=10\). That matches the graph's plateau height exactly.

Why the other options fail: Doubling the second hinge's weight (option with \(-2H_{20}\)) would overcorrect and make the function decrease again past \(S=20\), forming a triangle instead of a plateau. Flipping the overall sign (the \(-H_{10}+H_{20}\) option) would send the function negative, but the graph is non-negative. Shifting the first kink from \(S=10\) to \(S=15\) changes both the onset point and the plateau height (it would plateau at 5, not 10), so it doesn't fit either.

Only \(N(S)=H_{10}(S)-H_{20}(S)\) reproduces the exact onset at \(S=10\) and plateau at 10.

Correct option: (A)