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 initial equation is \(x \log x = 2 - x\). Reformatting the equation yields:

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

Step 2: Define the function \(f(x) = (x - 2) \log x\). The goal is to determine when \(f(x) = 2 - x\).

Step 3: The function is continuous and differentiable on \((1, 2)\). Through graphical or numerical analysis, at least one root is found within \((1, 2)\).

Step 4: Consequently, the equation possesses at least one root within the interval \((1, 2)\).