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MHT CET202521 Apr 2025Morning ShiftMathematicsApplication of DerivativesActual

The function f defined by f (x)=(x+2) e ^ -x is

Options

  1. Adecreasing for all x R
  2. Bdecreasing in (- ,-1) and increasing in (-1, )
  3. Cdecreasing in (-1, ) and increasing in (- ,-1)
  4. Dincreasing for all x R

Correct answer

C. decreasing in (-1, ) and increasing in (- ,-1)

Step-by-step solution

The derivative of f(x) = (x + 2)e^ -x is found using the product rule, yielding f'(x) = -e^ -x (x + 1) . Since e^ -x > 0 for all x , the sign of f'(x) depends on - (x + 1) . Setting f'(x) = 0 gives the critical point x = -1 . For x 0 , so f'(x) > 0 and f(x) is increasing. For x > -1 , -(x + 1) Thus, f(x) is increasing on (- , -1) and decreasing on (-1, ) . Matching the intervals, the correct choice is C .

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