Question

If \(y = f(x)\) is a monotonically increasing function such that \((\frac{dy}{dx})^2 = 6-\frac{dy}{dx}\) and \(y(0) = 5\), then \(y(3) = \cdots\)

• A. \(23\)
• B. \(14\)
• C. \(13\)
• D. \(11\)

Hint:

Solve the quadratic for dy/dx and pick the positive root for an increasing function.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Let \(p = \frac{dy}{dx}\). The equation \(p^2 = 6 - p\) is a quadratic in \(p\).

Step 2: Solve:
\(p^2 + p - 6 = 0\), so \((p + 3)(p - 2) = 0\), giving \(p = -3\) or \(p = 2\).

Step 3: Use monotonicity:
Since \(y\) is monotonically increasing, \(\frac{dy}{dx} \ge 0\), so \(p = 2\). Then \(y = 2x + c\). With \(y(0) = 5\), \(c = 5\), so \(y = 2x + 5\).

Step 4: Evaluate:
\(y(3) = 6 + 5 = 11\).

Step 5: Why the other options are wrong.
The value 23 would come from \(p = 6\), 14 from \(p = 3\), and 13 from \(p = \frac83\). Using \(p = -3\) would give \(y(3) = -4\), which is not an option and is excluded by monotonicity.

Final Answer:
\(y(3) = 11\), option (D). \[ \boxed{11} \]