Let \( f(x)=\frac{x^{2}}{4}(2 \ln x-1)-e x+k, k \subset R \). If least value of \( k \) for which \( \sqrt{f(x)} \) is defined for all \( x \in(0, \infty) \) is \( \frac{a}{b} e^{2} \), where \( a, b \in N \) then find the least value of \( (a t+b) \).

Question

Let \( f(x)=\frac{x^{2}}{4}(2 \ln x-1)-e x+k, k \subset R \). If least value of \( k \) for which \( \sqrt{f(x)} \) is defined for all \( x \in(0, \infty) \) is \( \frac{a}{b} e^{2} \), where \( a, b \in N \) then find the least value of \( (a +b) \).