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Paragraph: If a continuous f defined on the real line R , assume positive and negative values in R , then the equation f(x)=0 has a root in R . For example, if it is known that a continuous function f on R is positive at some point and its minimum values is negative, then the equation f(x)=0 has a root in R . Consider f(x)=k e^x-x for all real x , where k is real constant. Question: The positive value of k for which

Options

  1. A1 e
  2. B1
  3. Ce
  4. D_e 2

Correct answer

A. 1 e

Step-by-step solution

Let f(x)=k e^x-x aligned & & f^ (x) & =k e^x-1=0 & & x & =- k & & f^ (x) & =k e^x & Hence, & [f^ (x) ]_ x=- k & =1>0 & & f(- k) & =1+ k aligned For one root of given equation aligned 1+ k & =0 Hence, k & = 1 e . aligned

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