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 \(f'(x)\).
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 \(f'(x)\).
Updated On: Jan 31, 2023
- (I) (iii) (P)
- (II) (iv) (Q)
- (III) (i) (R)
- (II) (iii) (P)
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The Correct Option is C
Solution and Explanation
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) $f ^{\prime}(1)=1, f ^{\prime}( e )=\frac{1}{ e }-1<0$
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