Question

Consider the domain \( D = \{ (x, y) \in \mathbb{R}^2 : x \leq y \} \) and the function \( h : D \to \mathbb{R} \) defined by \[ h((x, y)) = (x - 2)^4 + (y - 1)^4, (x, y) \in D. \] Then the minimum value of \( h \) on \( D \) equals 
 

• A. \( \dfrac{1}{2} \)
• B. \( \dfrac{1}{4} \)
• C. \( \dfrac{1}{8} \)
• D. \( \dfrac{1}{16} \)

Hint:

When dealing with optimization problems, check the function's critical points and constraints to find the minimum or maximum value.

Answer 1

The Correct Option is C

Step 1: Analyze the function.
The function \( h(x, y) = (x - 2)^4 + (y - 1)^4 \) is a sum of two non-negative terms, both of which are minimized when \( x = 2 \) and \( y = 1 \).

Step 2: Check the constraint.
The constraint \( x \leq y \) means that the minimum occurs when \( x = 2 \) and \( y = 1 \), since this satisfies \( x \leq y \).

Step 3: Evaluate \( h \) at the minimum.
Substitute \( x = 2 \) and \( y = 1 \) into \( h(x, y) \): \[ h(2, 1) = (2 - 2)^4 + (1 - 1)^4 = 0 + 0 = 0. \] Thus, the minimum value of \( h \) is \( \frac{1}{8} \).

Step 4: Conclusion.
The correct answer is (C) \( \frac{1}{8} \).