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KCET2012MathematicsApplication of Derivatives

The maximum value of x e ^ - x is

Options

  1. Ae
  2. B1 e
  3. C- e
  4. D- 1 e

Correct answer

B. 1 e

Step-by-step solution

Let y=x e^ -x On differentiating w.r.t. ' x ', we get aligned & d y d x =x e^ -x (-1)+e^ -x & d y d x =e^ -x (1-x) aligned For maximum or minimum, dy dx =0 array lrr & e ^ - x (1- x )=0 & & 1- x =0 & ( e ^ - x 0 ) & x =1 & array From Eq. (ii), we get gathered aligned d ² y dx ² &= e ^ - x (-1)+(1- x ) e ^ - x (-1) &=- e ^ - x - e ^ - x + x e ^ - x &=-2 e ^ - x + xe ^ - x d ² y dx ² &= e ^ - x ( x -2) &= e ⁻¹(1-2)= 1 e (-1) ( d ² y dx )_ at x =1 aligned At x =1, y is maximum and maximum value =1 e ⁻¹= 1 e . gathered

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