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JEE Advance - Mathematics (2017 - Paper 1 Offline - No. 16)

By approximately matching the information given in the three columns of the following table.

Let f(x) = x + loge x $$-$$ x loge x, x$$ \in $$(0, $$\infty $$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

Column - 1 Column - 2 Column - 3
(i) f(x) = 0 for some $$x \in (1,{e^2})$$ (i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$ f is increasing in (0, 1)
(ii) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty $$ f is decreasing in (e, $${e^2}$$)
(iii) f'(x) = 0 for some $$x \in (0,1)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty $$ f' is increasing in (0, 1)
(iv) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$ f' is decreasing in (e, $${e^2}$$)
By approximately matching the information given in the three columns of the following table.

Let f(x) = x + loge x $$-$$ x loge x, x$$ \in $$(0, $$\infty $$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

Column - 1 Column - 2 Column - 3
(i) f(x) = 0 for some $$x \in (1,{e^2})$$ (i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$ f is increasing in (0, 1)
(ii) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty $$ f is decreasing in (e, $${e^2}$$)
(iii) f'(x) = 0 for some $$x \in (0,1)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty $$ f' is increasing in (0, 1)
(iv) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$ f' is decreasing in (e, $${e^2}$$)
By approximately matching the information given in the three columns of the following table.

Let f(x) = x + loge x $$-$$ x loge x, x$$ \in $$(0, $$\infty $$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

Column - 1 Column - 2 Column - 3
(i) f(x) = 0 for some $$x \in (1,{e^2})$$ (i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$ f is increasing in (0, 1)
(ii) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty $$ f is decreasing in (e, $${e^2}$$)
(iii) f'(x) = 0 for some $$x \in (0,1)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty $$ f' is increasing in (0, 1)
(iv) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$ f' is decreasing in (e, $${e^2}$$)
Which of the following options is the only INCORRECT combination?
(I) (iii) (P)
(II) (iv) (Q)
(II) (ii) (P)
(III) (i) (R)

Explanation

$$f(x) = x + \ln x - x\ln x$$

$$f(1) = 1 > 0$$


$$f({e^2}) = {e^2} + 2 - 2{e^2} = 2 - {e^2} < 0$$

$$ \Rightarrow f(x) = 0$$ for some $$x \in (1,\,{e^2})$$

$$ \therefore $$ I is correct.

$$f'(x) = 1 + {1 \over x} - \ln x - 1$$

$$ = {1 \over x} - \ln x$$

$$f'(x) > 0$$ for (0, 1)

$$f'(x) < 0$$ for $$(e,\infty )$$

$$ \therefore $$ P and Q are correct, II is correct, III is incorrect.

$$f''(x) = {{ - 1} \over {{x^2}}} - {1 \over x}$$

$$f''(x) < 0$$ for $$(0,\infty )$$

$$ \therefore $$ S, is correct, R is incorrect.

IV is incorrect.

$$\mathop {\lim }\limits_{x \to \infty } f(x) = - \infty $$

$$\mathop {\lim }\limits_{x \to \infty } f'(x) = - \infty $$

$$\mathop {\lim }\limits_{x \to \infty } f''(x) = 0$$

$$ \therefore $$ ii, iii, iv are correct.

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