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The curve y = f(x), representing a parabola, is tangent to the line y = x at the point where x = 1. Which of the following is true?

1. f'(0) equals f''(1)
2. f'(1) is equal to 1
3. f(0) plus f''(0) plus f'(0) equals 1
4. Twice f(0) is equal to 1 minus f'(0)

Correct answer: Twice f(0) is equal to 1 minus f'(0)

Solution

For parabola f(x)=Ax^2+Bx+C tangent to y=x at x=1: f(1)=A+B+C=1 and f'(1)=2A+B=1, giving C=A and B=1-2A. Then 2f(0)=2C=2A and 1-f'(0)=1-B=2A, so 2f(0)=1-f'(0) holds identically (idx 3). Stored idx 2 (f(0)+f''(0)+f'(0)=1) reduces to A+1, true only if A=0 (not a parabola).

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