Question

Let \(f(x)=x+log_{e}​x−xlog_{e}​x,\text{ }x∈(0,∞)\).

• Column 1 contains information about zeros of \(f(x)\), \(f'(x)\) and \(f''(x)\).
• Column 2 contains information about the limiting behavior of \(f(x)\), \(f'(x)\) and \(f''(x)\) at infinity.
• Column 3 contains information about increasing/decreasing nature of \(f(x)\) and \(f'(x)\).

• A. (I) (iii) (P)
• B. (II) (iv) (Q)
• C. (III) (i) (R)
• D. (II) (iii) (P)

Answer 1

The Correct Option is C

1) $f ( x )= x +\log _{e}( x )- x \log _{e}( x )$
2) $f ^{\prime}( x )=\frac{1}{ x }-\log _{ e }( x )$
3) $f^{\prime \prime}(x)=-\frac{(x+1)}{x^{2}}<0 \forall x>0$
4) $f (1)= f ( e )=1, f \left( e ^{2}\right)<0$
5) $f ^{\prime}(1)=1, f ^{\prime}( e )=\frac{1}{ e }-1<0$