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

Step 1: Understand the building block \(H_K(S) = \max(S-K, 0)\). This is a "ramp" function: it equals 0 for \(S \le K\), and rises with slope 1 for \(S > K\). This is exactly the shape of a shifted ReLU with kink at \(S=K\).

Step 2: Read the shape of \(N(S)\) from the given graph. The graph shows \(N(S)=0\) for \(S \le 10\), then \(N(S)\) rises linearly with slope 1 starting at \(S=10\), reaching the value 10 at \(S=20\), and then stays flat (constant) at 10 for all \(S > 20\). This is a ramp that saturates - it turns on at \(S=10\) and turns off its slope at \(S=20\).

Step 3: Test option (A): \(N(S) = H_{10}(S) - H_{20}(S)\).
For \(S \le 10\): both terms are 0, so \(N(S)=0\). ✓
For \(10 < S \le 20\): \(H_{10}(S) = S-10\), \(H_{20}(S)=0\), so \(N(S) = S - 10\), rising from 0 to 10. ✓
For \(S > 20\): \(H_{10}(S) = S-10\), \(H_{20}(S) = S-20\), so \(N(S) = (S-10)-(S-20) = 10\), a constant plateau. ✓
This exactly matches the graph.

Step 4: Rule out the others using a test point, say \(S=25\).
Option (B): \(H_{10}(25) - 2H_{20}(25) = 15 - 2(5) = 5\), not 10 - fails, since (B) actually decreases back toward zero after \(S=20\) (a triangular "tent", not a plateau).
Option (C): \(-H_{10}(S)+H_{20}(S)\) is simply the negative of option (A), giving negative values where the graph is positive - fails.
Option (D): \(H_{15}(S)-H_{20}(S)\) only starts rising at \(S=15\) (not 10) and plateaus at value 5 (not 10) - fails to match the graph's onset point and plateau height.

\[ N(S) = H_{10}(S) - H_{20}(S) \]

Final Answer: (A) \(H_{10}(S) - H_{20}(S)\)