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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2024 · Mathematics · Application of Derivatives

Let f(x)=(x)/( x) ;(x > 1) Consider the following statements: 1....

Let f(x)=xln⁡x;(x>1)f(x)=\frac{x}{\ln x} ;(x > 1) Consider the following statements: 1. f(x)f(x) is increasing in the interval (e,∞)2.(e, \infty) 2. f(x)isdecreasingintheintervalis decreasing in the interval(1, e)3.9ln⁡7>7ln⁡93. 9 \ln 7 > 7 \ln 9 Which of the statements given above are correct?

Options

  1. A1 and 2 only
  2. B2 and 3 only
  3. C1 and 3 only
  4. D1, 2 and 3

Correct answer

D. 1, 2 and 3

Step-by-step solution

f(x)=xln⁡(x)(x>1)f(x)=ln⁡x−1[ln⁡(x)]2\begin{aligned} & f(x)=\frac{x}{\ln (x)}(x > 1) \\ & f(x)=\frac{\ln x-1}{[\ln (x)]^2}\end{aligned}
f(x)f(x) is +ve,x∈(e,∞)⇒+\mathrm{ve}, x \in(e, \infty) \Rightarrow (1) is correct f(x)f(x) is -ve ∀x∈(1,e)⇒\forall x \in(1, e) \Rightarrow (2) is correct As ln⁡x\ln x is an increasing function 79>97ln⁡79>ln⁡979ln⁡7>7ln⁡9\begin{aligned} 7^9 & > 9^7 \\ \ln 7^9 & > \ln 9^7 \\ 9 \ln 7 & > 7 \ln 9 \end{aligned}\Rightarrow$ statement (3) is correct Hence, statements (1), (2) and (3) are correct.
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