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Let f(x) =x+logex-xlogex ,x(0,oo)dot Co...

Let `f(x)` `=x+log_ex-xlog_ex ,x(0,oo)dot` Column 1 contains information about zeros of `f^(prime)(x)f^(prime)(x)a n df^(x)dot` Column 2 contains information about the limiting behaviour of `f^(prime)(x)f^(prime)(x)a n df^(x)` at infinity. Column 2 contains information about the increasing/decreasing nature of `f(x)a n df^(prime)(x)dot` Column I, Column 2, Column 3 I, `f(x)=0forsom ex(l , e^2)` , (i), `("lim")_("x"vecoo"")f^(prime)(x)=0` , (P), `f` is increasing in (0,1) II, `f'(x)=0forsom ex(l , e)` , (ii), `("lim")_("x"vecoo"")f^(x)=-oo` , (Q), `f` is decreasing in `(e ,e^2)` III, `f'(x)=0forsom ex(0,1)` , , `("lim")_("x"vecoo"")f^(prime)(x)=-oo` , (R), `f` is increasing in (0,1) IV, `f^(' '(x))=0forsom ex(1, e)` , , `("lim")_("x"vecoo"")f^prime^'(x)=0` , (S), `f` is decreasing in (`e , e^2` ) Which of the following options is the only CORRECT combination? (I) (ii) (P) (b) (IV) (iv) (S) (III) (iii) (R) (d) (II) (ii) (Q) Which of the following option is the only incorrect combination? (III) (i) (R) (b) (I) (iii) (P) (II) (iii) (P) (d) (II) (iv) (Q) Which of the following options is the only CORRECT combination? (I) (ii) (R) (b) (II) (iii) (S) (III) (iv) (P) (d) (IV) (i) (S)

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