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

Answer: (C)

(III) (i) (R)

Answer 2

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. $f ^{\prime}(1)=1, f ^{\prime}( e )=\frac{1}{ e }-1<0$