Questions: Find the open intervals where the function is concave upward or concave downward. Find any inflection points. f(x)=14 x=14 e^(-x) Where is the function concave upward and where is it concave downward? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function is concave upward on the interval(s) . The function is never concave downward. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The function is concave upward on the interval(s) and concave downward on the interval(s) 1. (Type your answers in interval notation. Use integers or fractions for any numbers in the expressions. Use a comma to separate answers as needed.) C. The function is concave downward on the interval(s) (-∞, ∞). The function is never concave upward. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) D. The function is never concave upward or downward. Find any inflection points of f. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function has an inflection point at . (Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) B. The function f has no inflection points.

Find the open intervals where the function is concave upward or concave downward. Find any inflection points.
f(x)=14 x=14 e^(-x)

Where is the function concave upward and where is it concave downward? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The function is concave upward on the interval(s) . The function is never concave downward.
(Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
B. The function is concave upward on the interval(s)  and concave downward on the interval(s) 1.
(Type your answers in interval notation. Use integers or fractions for any numbers in the expressions. Use a comma to separate answers as needed.)
C. The function is concave downward on the interval(s) (-∞, ∞). The function is never concave upward.
(Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
D. The function is never concave upward or downward.

Find any inflection points of f. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The function has an inflection point at .
(Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.)
B. The function f has no inflection points.
Transcript text: Find the open intervals where the function is concave upward or concave downward. Find any inflection points. \[ f(x)=14 x=14 e^{-x} \] Where is the function concave upward and where is it concave downward? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function is concave upward on the interval(s) $\square$ The function is never concave downward. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The function is concave upward on the interval(s) $\square$ and concave downward on the interval(s) $\square$ 1. (Type your answers in interval notation. Use integers or fractions for any numbers in the expressions. Use a comma to separate answers as needed.) C. The function is concave downward on the interval(s) $(-\infty, \infty)$. The function is never concave upward. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) D. The function is never concave upward or downward. Find any inflection points of f. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function has an inflection point at $\square$ . (Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) B. The function $f$ has no inflection points.
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Solution

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Solution Steps

Step 1: Find the First and Second Derivatives

The function is given by \( f(x) = 14e^{-x} \). The first derivative is calculated as follows: \[ f'(x) = -14e^{-x} \] The second derivative is: \[ f''(x) = 14e^{-x} \]

Step 2: Analyze the Second Derivative

The second derivative \( f''(x) = 14e^{-x} \) is always positive for all \( x \) since \( e^{-x} > 0 \) for all real numbers. This indicates that the function is concave upward everywhere.

Step 3: Determine Concavity and Inflection Points

Since \( f''(x) > 0 \) for all \( x \), the function is never concave downward. Additionally, there are no points where the second derivative changes sign, which means there are no inflection points.

Final Answer

A. The function is concave upward on the interval(s) \((-∞, ∞)\). The function is never concave downward.
B. The function has no inflection points.

Thus, the final answers are: \[ \boxed{\text{A: } (-\infty, \infty) \text{ and B: no inflection points}} \]

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