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MHT CET2007MathematicsApplication of Derivatives

On the interval [0,1] the function x²⁵(1-x)⁷⁵ takes its maximum value at the point

Options

  1. A0
  2. B1 / 4
  3. C1 / 2
  4. D1 / 3

Correct answer

B. 1 / 4

Step-by-step solution

Given, f(x)=x²⁵(1-x)⁷⁵ aligned f^ (x) &=25 x²⁴(1-x)⁷⁵-75 x²⁵(1-x)⁷⁴ &=25 x²⁴(1-x)⁷⁴(1-4 x) f^ (x) &=0 x &=0,1,1 / 4 aligned If x 0 . and if x>1 / 4 , then f^ (x)=25 x²⁴(1-x)⁷⁴(1-4 x) < 0 . Thus, f^ (x) changes its sign from positive to negative as x passes through 1 / 4 from left to right. Hence, f(x) attains its maximum at x=1 / 4 .

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