Find the smallest value of the function y=11x - In(x+15) 11 on the segment [-14,5;0]

To find the smallest value of the function \( y = 11x - \ln(x + 15) + 11 \) on the interval \( x \in [-14.5, 0] \), we need to analyze the function's behavior over this domain. Here are the steps to find the minimum:

### Step 1: Understand the Function
The function is given by:
\[
y(x) = 11x - \ln(x + 15) + 11
\]

1. **Domain**: The natural logarithm function \( \ln(x + 15) \) is defined when \( x + 15 > 0 \). Since the interval is \( x \in [-14.5, 0] \), all values of \( x + 15 \) are positive, and the function is well-defined over this interval.

### Step 2: Find the Derivative of the Function
To find critical points, we need to differentiate \( y(x) \) with respect to \( x \):

\[
y'(x) = \frac{d}{dx} (11x - \ln(x + 15) + 11)
\]

Using the derivative rules:
\[
y'(x) = 11 - \frac{1}{x + 15}
\]

### Step 3: Set the Derivative to Zero to Find Critical Points
Set \( y'(x) = 0 \):

\[
11 - \frac{1}{x + 15} = 0
\]

Solve for \( x \):

\[
11 = \frac{1}{x + 15}
\]

\[
x + 15 = \frac{1}{11}
\]

\[
x = \frac{1}{11} - 15
\]

\[
x = \frac{1 - 165}{11} = \frac{-164}{11}
\]

\[
x \approx -14.909
\]

This value is approximately \(-14.909\), which is outside the interval \([-14.5, 0]\). So, we need to check the endpoints of the interval to find the minimum value.

### Step 4: Evaluate the Function at the Endpoints
Evaluate \( y(x) \) at \( x = -14.5 \) and \( x = 0 \):

1. **At \( x = -14.5 \):**
   \[
   y(-14.5) = 11(-14.5) - \ln(-14.5 + 15) + 11
   \]
   \[
   y(-14.5) = -159.5 - \ln(0.5) + 11
   \]
   \[
   y(-14.5) = -159.5 - (-0.693) + 11
   \]
   \[
   y(-14.5) = -159.5 + 0.693 + 11
   \]
   \[
   y(-14.5) \approx -147.807
   \]

2. **At \( x = 0 \):**
   \[
   y(0) = 11(0) - \ln(0 + 15) + 11
   \]
   \[
   y(0) = 0 - \ln(15) + 11
   \]
   \[
   y(0) = -2.708 + 11
   \]
   \[
   y(0) \approx 8.292
   \]

### Step 5: Conclusion
The smallest value of \( y(x) \) on the interval \( [-14.5, 0] \) occurs at \( x = -14.5 \) and is approximately \( -147.807 \).

Would you like more details on the steps, or do you have any other questions?

### Related Questions:
1. How do we find critical points for a function in different types of intervals?
2. What is the significance of the derivative of a function in determining its minima and maxima?
3. How do logarithmic transformations affect the shape of a function?
4. How do we handle endpoints when finding the extreme values of a function?
5. What methods exist for finding minima and maxima in multivariable functions?

**Tip**: Always check the domain of a function before proceeding with differentiation to avoid undefined values!

Find the smallest value of the function y = 11x - ln(x + 15) + 11 on the segment [-14.5, 0]

Learn how to find the smallest value of the function y = 11x - ln(x + 15) + 11 on the segment [-14.5, 0] using differentiation and logarithmic functions. This problem demonstrates the application of the first derivative test and critical point analysis. Suitable for advanced high school or undergraduate students.

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