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AP EAMCET202319 May 2023Morning ShiftMathematicsApplication of DerivativesActual

In the interval ( 1 e , e ) , a decreasing function among the following functions is

Options

  1. Af(x)= x x
  2. Bf(x)=x^2 x
  3. Cf(x)=x x
  4. Df(x)=x^ -x

Correct answer

D. f(x)=x^ -x

Step-by-step solution

The best way to approach this question is, find 1^ st derivative of all the function at x ( 1 e , e ) you can choose x= 2 e , 3 e ( 1 e , e ) If f^ (x) is -ve at any point in the given interval then that f(x) is decreasing For option (a): f(x)= (x) x f^ (x)= 1- x x^2 . when x= 2 e ( 1 e , e ) f^ ( 2 e )>0 Hence f(x) is not decreasing. For option (b) : f(x)=x^2 (x)f^ (x)>0 when x ( 1 e , e ) For option (c): f(x)=x xf^ (x)>0 when x ( 1 e , e ) For option (d) : f(x)=x⁻ x f(x)=-x (x) ...(i) Hence f^ (x)=- x -x[1+ x] ..

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