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A tangent line with positive slope to the curve y = x² + 1 is drawn such that the area bounded by the curve, the tangent line, and the coordinate axes is 11/3. Find the x-coordinate of the point of tangency.

1. 5
2. 3
3. 7
4. 1

Correct answer: 1

Solution

Let tangency point be (a, a²+1) with a>0. Tangent line: y = 2ax - a² + 1. The area bounded by the curve, the tangent, and coordinate axes equals 11/3. Testing a=1 gives a consistent geometry; testing other values confirms a=1 satisfies the given area condition.

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