Question

Consider the function f(x) = (x−2)logx. Then the equation xlogx = 2−x has:

• A. at least one root in (1,2)
• B. has no root in (1,2)
• C. is not solvable
• D. has infinitely many roots in (−2, 1)

Hint:

When solving transcendental equations involving logarithmic terms, use numerical methods or graphical analysis to determine the roots

Answer 1

The Correct Option is A

Step 1: The equation is \(x \log x = 2 - x\). Rearranging the equation:

\[ x \log x + x - 2 = 0 \]

Step 2: Consider the function \(f(x) = (x - 2) \log x\). We need to analyze when the equation \(f(x) = 2 - x\) holds true.

Step 3: The function is continuous and differentiable in the interval \((1, 2)\), and using graphical or numerical methods, we find that there is at least one root in the interval \((1, 2)\).

Step 4: Therefore, the equation has at least one root in the interval \((1, 2)\).