Find the critical point(s) of the function 𝑓(𝑥)=(9−13𝑥)/ (13^𝑥)
, if any, leaving your answer(s) in exact form (no approximations). Enter DNE if there are no critical points.

To find the critical points of the function \( f(x) = \frac{9 - 13x}{13^x} \), we first need to compute its derivative \( f'(x) \) and then set it equal to zero to find any critical points.

### Step 1: Compute the derivative \( f'(x) \)

Using the quotient rule, which states that for a function \( g(x) = \frac{u(x)}{v(x)} \), its derivative is given by:
\[
g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}
\]
we'll apply this to \( f(x) = \frac{9 - 13x}{13^x} \).

Here, \( u(x) = 9 - 13x \) and \( v(x) = 13^x \). We need to find the derivatives \( u'(x) \) and \( v'(x) \):
- \( u'(x) = -13 \)
- \( v(x) = 13^x \) and \( v'(x) = 13^x \ln(13) \) (since the derivative of \( a^x \) is \( a^x \ln(a) \)).

Now apply the quotient rule:
\[
f'(x) = \frac{(-13)(13^x) - (9 - 13x)(13^x \ln(13))}{(13^x)^2}
\]
Simplifying:
\[
f'(x) = \frac{-13 \cdot 13^x - (9 - 13x) \cdot 13^x \ln(13)}{(13^x)^2}
\]
Factoring out \( 13^x \) in the numerator:
\[
f'(x) = \frac{13^x \left[-13 - (9 - 13x)\ln(13)\right]}{(13^x)^2}
\]
Cancel \( 13^x \) in the numerator and denominator:
\[
f'(x) = \frac{-13 - (9 - 13x)\ln(13)}{13^x}
\]

### Step 2: Set \( f'(x) = 0 \) to find critical points
We set the numerator equal to zero to solve for \( x \):
\[
-13 - (9 - 13x)\ln(13) = 0
\]
Solve for \( x \):
\[
(9 - 13x)\ln(13) = -13
\]
\[
9 - 13x = \frac{-13}{\ln(13)}
\]
\[
-13x = \frac{-13}{\ln(13)} - 9
\]
\[
x = \frac{\frac{13}{\ln(13)} + 9}{13}
\]

This is the exact form of the critical point.

### Final Answer:
The critical point is \( x = \frac{\frac{13}{\ln(13)} + 9}{13} \).

Would you like more details or have any questions?

Here are some related questions to explore:
1. How do you classify a critical point as a maximum, minimum, or saddle point?
2. What role does the second derivative play in determining the nature of a critical point?
3. Can this function have multiple critical points? Why or why not?
4. What happens to the function's behavior as \( x \to \infty \)?
5. How does changing the base of the exponent (e.g., using 14 instead of 13) affect the critical points?

**Tip:** The quotient rule is essential for differentiating ratios of functions efficiently. Always simplify after applying it to avoid complicated expressions.

Find the critical point(s) of the function 𝑓(𝑥)=(9−13𝑥)/ (13^𝑥), if any, leaving your answer(s) in exact form (no approximations). Enter DNE if there are no critical points.

Explore how to find the critical points of the function f(x) = (9 - 13x) / (13^x) using the quotient rule. This problem involves advanced calculus concepts, including the differentiation of exponential functions and the application of the quotient rule. Suitable for college-level calculus students.

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