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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2023 · Mathematics · Application of Derivatives

Consider the following statements: 1. f(x)= x is increasing in (0,...

Consider the following statements: 1. f(x)=ln⁡xf(x)=\ln x is increasing in (0,∞)(0, \infty) 2. g(x)=ex+e1xg(x)=e^x+e^{\frac{1}{x}} is decreasing in (0,∞)(0, \infty) Which of the statements given above is/are correct?

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

A. 1 only

Step-by-step solution

f(x)=ln⁡x⇒f′(x)=1xf(x)=\ln x \Rightarrow f^{\prime}(x)=\frac{1}{x} for x∈(0,∞),f′(x)>0∴f(x)x \in(0, \infty), f^{\prime}(x) > 0\therefore f(x) is increasing in (0,∞)g(x)=ex+e1/xg′(x)=ex−e1/x(1x2)when(0, \infty)g(x)=e^x+e^{1 / x}g^{\prime}(x)=e^x-e^{1 / x} \left(\frac{1}{x^2}\right) when x < 1, g^{\prime}(x) < 0atatx=1, g^{\prime}(x)=0atatx > 1, g^{\prime}(x) > 0\therefore \quadinin(0, ∞),\infty), g(x)isneithercontinuouslyincreasingnordecreasing.Only1iscorrect.is neither continuously increasing nor decreasing. Only 1 is correct.
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