Questions: Answer the following questions about the function whose derivative is given. a. What are the critical points of f ? b. On what intervals is f increasing or decreasing? c. At what points, if any, does f assume local maximum and minimum values? f'=(x-7)e^(-x) a. Select the correct choice below and fill in any answer boxes in your choice. A. The critical point(s) is/are x= . (Simplify your answer. Use a comma to separate answers as needed.) B. There are no critical points. b. Select the correct choice below and fill in any answer boxes in your choice. A. The function f(x) is increasing on and decreasing on . (Simplify your answer. Use interval notation. Use a comma to separate answers as needed.) B. The function is always increasing or always decreasing. c. Select the correct choice below and fill in any answer boxes in your choice. A. There are no local minima, but there is a local maximum at a critical point. B. There are no local maxima, but there is a local minimum at a critical point.

Answer the following questions about the function whose derivative is given. a. What are the critical points of f ? b. On what intervals is f increasing or decreasing? c. At what points, if any, does f assume local maximum and minimum values? f'=(x-7)e^(-x) a. Select the correct choice below and fill in any answer boxes in your choice. A. The critical point(s) is/are x= . (Simplify your answer. Use a comma to separate answers as needed.) B. There are no critical points. b. Select the correct choice below and fill in any answer boxes in your choice. A. The function f(x) is increasing on and decreasing on . (Simplify your answer. Use interval notation. Use a comma to separate answers as needed.) B. The function is always increasing or always decreasing. c. Select the correct choice below and fill in any answer boxes in your choice. A. There are no local minima, but there is a local maximum at a critical point. B. There are no local maxima, but there is a local minimum at a critical point.
Transcript text: Answer the following questions about the function whose derivative is given. a. What are the critical points of $f$ ? b. On what intervals is $f$ increasing or decreasing? c. At what points, if any, does $f$ assume local maximum and minimum values? \[ f^{\prime}=(x-7) e^{-x} \] a. Select the correct choice below and fill in any answer boxes in your choice. A. The critical point(s, is/are $x=$ $\square$ $\square$. (Simplify your answer. Use a comma to separate answers as needed.) B. There are no critical points. b. Select the correct choice below and fill in any answer boxes in your choice. A. The function $f(x)$ is increasing on $\square$ and decreasing on $\square$ . (Simplify your answer. Use interval notation. Use a comma to separate answers as needed.) B. The function is always increasing or always decreasing. c. Select the correct choice below and fill in any answer boxes in your choice. A. There are no local minima, but there is a local maximum at a critical point. B. There are no local maxima, but there is a local minimum at a critical point.
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Solution

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Solution Steps

Step 1: Critical Points

To find the critical points of the function \( f \), we set the derivative \( f' = (x - 7)e^{-x} \) to zero. This gives us the critical point: \[ x = 7 \]

Step 2: Intervals of Increase and Decrease

Next, we analyze the sign of the derivative \( f' \) to determine where the function is increasing or decreasing. The derivative is positive for \( x > 7 \) and negative for \( x < 7 \). Therefore, we conclude:

  • \( f \) is increasing on the interval \( (7, \infty) \).
  • \( f \) is decreasing on the interval \( (-\infty, 7) \).
Step 3: Local Maxima and Minima

Finally, we evaluate the critical point \( x = 7 \) to determine if it is a local maximum or minimum. Since \( f' \) changes from negative to positive at \( x = 7 \), we find that:

  • There is a local minimum at \( x = 7 \).

Final Answer

The answers to the questions are as follows:

  • a. The critical point(s) is/are \( x = 7 \).
  • b. The function \( f(x) \) is increasing on \( (7, \infty) \) and decreasing on \( (-\infty, 7) \).
  • c. There are no local maxima, but there is a local minimum at a critical point.

Thus, the final answers are: \[ \boxed{x = 7} \] \[ \text{Increasing: } (7, \infty), \text{ Decreasing: } (-\infty, 7) \] \[ \text{Local Minimum: } x = 7 \]

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