Question

Let \(A(1,15),\ B(3,-12),\ C(6,12)\) be three consecutive turning points of a continuous curve \(y = f(x)\). If \(f(x) = 0\) only for \(x = \alpha\) and \(x = \beta\), then \[ |\beta - \alpha| < ? \]

• A. \(27\)
• B. \(2\)
• C. \(5\)
• D. \(24\)

Hint:

To estimate roots of a function between known turning points, use sign changes and continuity of the curve.

Answer 1

The Correct Option is C

Given points are turning points: A(1,15), B(3,-12), C(6,12). These suggest the shape of the function is oscillatory, and the curve must cross the x-axis between turning points. We are told that \(f(x) = 0\) only at two points \(\alpha\) and \(\beta\). From the curve behavior, the only interval between turning points where the curve changes sign is between A and B, and B and C. So \(\alpha\) and \(\beta\) must lie in \((1, 6)\), i.e., \(|\beta - \alpha|<5\)