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AP EAMCET202317 May 2023Evening ShiftMathematicsApplication of DerivativesActual

An extreme value of f(x)= 4 x + 1 1- x in (0, / 2) is

Options

  1. A9
  2. B8
  3. C2 / 3
  4. D-7 / 2

Correct answer

A. 9

Step-by-step solution

f(x)= 4 x + 1 1- x f^ (x)=- 4 x ^2 x + x (1- x)^2 For critical points: f^ (x)=0 aligned & - 4 x ^2 x + x (1- x)^2 =0 & - x [ 4(1- x)^2+ ^2 x ^2 x(1- x)^2 ]=0 & 4(1- x)^2+ ^2 x=0 & 3 ^3 x-8 x+4=0 & (3 x-2)( x-2)=0 & 3 x-2=0 or x=2( not possible) & x= 2 3 aligned So, the only critical value is x= ⁻¹ ( 2 3 ) f(x) has extremum at x= ⁻¹ ( 2 3 ) So, the extreme value is aligned & f ( ⁻¹ 2 3 )= 4 ( ⁻¹ 2 3 ) + 1 1- ( ⁻¹ 2 3 ) & =4 3 2 +3=9 aligned

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