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AP EAMCET2007MathematicsApplication of Derivatives

Observe the statements given below : Assertion (A) : f(x)=x e^ -x has the maximum at x=1 Reason (R) : f^ (1)=0 and f^ (1) < 0 Which of the following is correct?

Options

  1. ABoth (A) and (R) are true and (R) is the correct reason for (A)
  2. BBoth (A) and (R) are true, but (R) is not the correct reason for (A)
  3. C(A) is true, (R) is false
  4. D(A) is false, (R) is true

Correct answer

A. Both (A) and (R) are true and (R) is the correct reason for (A)

Step-by-step solution

Given, f(x)=x e^ -x aligned f^ (x) & =e^ -x -x e^ -x f^ (x) & =-e^ -x -e^ -x +x e^ -x & =-2 e^ -x +x e^ -x aligned For maximum, put f^ (1)=0 x=1 and f^ (1)=-1 < 0 Both A and R are true and R is the correct reason for A .

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