Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable
Function =(\boldsymbol{x}+\mathbf{2}) \boldsymbol{e}^{-\boldsymbol{x}} \) is
Function \( \boldsymbol{f}(\boldsymbol{x})=(\boldsymbol{x}+\mathbf{2}) \boldsymbol{e}^{-\boldsymbol{x}} \) is
- A. decreasing
- B. decreasing in \( (-\infty,-1) \) and increasing in \( (-1, \infty) \)
- C. increasing
- D. decreasing in \( (-1, \infty) \) and increasing in \( (-\infty,-1) \)
Step-by-step solution
First, find the derivative: f'(x) = (1)e^{-x} + (x+2)(-e^{-x}) = e^{-x}(1 - x - 2) = -(x+1)e^{-x}. Since e^{-x}>0, the sign of f'(x) is opposite to (x+1). f'(x)>0 when x<-1 (increasing) and f'(x)<0 when x>-1 (decreasing). Hence, f is increasing on (-∞,-1) and decreasing on (-1,∞).
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