The root of the equation \(xe^{x} = 1\) between 0 and 1, obtained by using two iterations of bisection method, is
GATE Mathematics - 2012,
Answer: C
Two iterations of bisection give 0.75.
Let \(f(x) = xe^{x} - 1\). Then \(f(0) = -1 \lt 0\) and \(f(1) = e - 1 \gt 0\), so a root lies in [0, 1].
\[ \begin{aligned} x_{1} &= 0.5:\ f(0.5) = 0.5e^{0.5} - 1 = -0.176 \lt 0 \\ x_{2} &= \tfrac{0.5 + 1}{2} = 0.75 \end{aligned} \]
Since \(f(0.5) \lt 0\), the root lies in [0.5, 1], and the second midpoint is 0.75. (The true root is about 0.567.) 0.25 would be the second midpoint only if \(f(0.5)\) were positive; 0.50 is the first iterate, and 0.65 is not a bisection midpoint of [0, 1].
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