Prove that for `lambda gt 1`, the equation `xlog x +x =lambda` has least one solution in `[1 , lambda]`.

Explanation

To prove that for \( \lambda > 1 \), the equation \( x \log x + x = \lambda \) has at least one solution in the interval \([1, \lambda]\), we can follow these steps:

### Step 1: Define the function
Let \( f(x) = x \log x + x - \lambda \). We want to show that \( f(x) = 0 \) has at least one solution in the interval \([1, \lambda]\).

### Step 2: Check continuity
The function \( f(x) \) is continuous on the interval \([1, \lambda]\) because it is composed of continuous functions (the logarithm function and polynomial functions).

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