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AP EAMCET202124 Aug 2021Morning ShiftMathematicsApplication of DerivativesActual

If f(x)=x^ x and f(0)=0 , then the value of for which Rolle's theorem can be applied in [0,1] is

Options

  1. A-2
  2. B-1
  3. C0
  4. D1 / 2

Correct answer

D. 1 / 2

Step-by-step solution

f(x)=x^a x and f(0)=0 Rolle's theorem is applicable in [0,1] . It is continuous in closed interval [0,1] and It is differentiable in (a, b) and f(0)=f(1) . f(x) is continuous at x=0 _ x 0⁺ f(x)=f(0) _ x 0⁺ x^a x=0 aligned _ x 0⁺ x x^ -a & =0 & [ It is indeterminate form ] aligned Using L' hospital rule _ x 0⁺ 1 x -a x^ -a-1 =0 _ x 0⁺ - x^a a =0 It is possible when a>0 a= 1 2

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