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← JEE Advanced 2017 · Paper 1
MathematicsSingle correct+3 / -1Question 53 of 54

JEE Advanced 2017 · Paper 1, Mathematics question 17

Passage

Answer Q.52, Q.53 and Q.54 by appropriately matching the information given in the three columns of the following table.

Let f(x)=x+log⁡ex−xlog⁡exf(x) = x + \log_e x - x\log_e x, x∈(0,∞)x \in (0, \infty).

  • Column 1 contains information about zeros of f(x)f(x), f′(x)f'(x) and f′′(x)f''(x).
  • Column 2 contains information about the limiting behavior of f(x)f(x), f′(x)f'(x) and f′′(x)f''(x) at infinity.
  • Column 3 contains information about increasing/decreasing nature of f(x)f(x) and f′(x)f'(x).
Column 1 Column 2 Column 3
(I) f(x)=0f(x) = 0 for some x∈(1,e2)x \in (1, e^2) (i) lim⁡x→∞f(x)=0\lim_{x\to\infty} f(x) = 0 (P) ff is increasing in (0,1)(0, 1)
(II) f′(x)=0f'(x) = 0 for some x∈(1,e)x \in (1, e) (ii) lim⁡x→∞f(x)=−∞\lim_{x\to\infty} f(x) = -\infty (Q) ff is decreasing in (e,e2)(e, e^2)
(III) f′(x)=0f'(x) = 0 for some x∈(0,1)x \in (0, 1) (iii) lim⁡x→∞f′(x)=−∞\lim_{x\to\infty} f'(x) = -\infty (R) f′f' is increasing in (0,1)(0, 1)
(IV) f′′(x)=0f''(x) = 0 for some x∈(1,e)x \in (1, e) (iv) lim⁡x→∞f′′(x)=0\lim_{x\to\infty} f''(x) = 0 (S) f′f' is decreasing in (e,e2)(e, e^2)

Which of the following options is the only CORRECT combination?

Choose one option
Answer and worked solution

Official answer: B

Using the facts from the analysis of f(x)=x+ln⁡x−xln⁡xf(x)=x+\ln x-x\ln x: f′=1x−ln⁡xf'=\tfrac1x-\ln x is decreasing, has its zero in (1,e)(1,e), and tends to −∞-\infty.

(II)(iii)(S) is all true: f′=0f'=0 in (1,e)(1,e), f′→−∞f'\to-\infty, and f′f' is decreasing on (e,e2)(e,e^2).

The others fail: (R) is false; (III) is false; (IV) is false.

(B)

Topics:MonotonicityLimits
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