If `f(x)=" max "(4x+3, 3x+6)" for "x in [-6, 10],` Find the maximum value of `f(x)`.

Answer

To find the maximum value of the function \( f(x) = \max(4x + 3, 3x + 6) \) for \( x \) in the interval \([-6, 10]\), we will follow these steps:

### Step 1: Identify the two functions
We have two linear functions:
- \( y_1 = 4x + 3 \)
- \( y_2 = 3x + 6 \)

### Step 2: Find the intersection point of the two functions
To find where the two functions intersect, we set them equal to each other:
\[ 4x + 3 = 3x + 6 \]
Subtract \( 3x \) from both sides:
\[ 4x - 3x + 3 = 6 \]
This simplifies to:
\[ x + 3 = 6 \]
Subtract 3 from both sides:
\[ x = 3 \]

### Step 3: Calculate the function values at the intersection point
Now we will find the value of \( f(x) \) at \( x = 3 \):
\[ f(3) = \max(4(3) + 3, 3(3) + 6) = \max(12 + 3, 9 + 6) = \max(15, 15) = 15 \]

### Step 4: Evaluate the function at the endpoints of the interval
Next, we evaluate \( f(x) \) at the endpoints of the interval, \( x = -6 \) and \( x = 10 \).

1. For \( x = -6 \):
\[ f(-6) = \max(4(-6) + 3, 3(-6) + 6) = \max(-24 + 3, -18 + 6) = \max(-21, -12) = -12 \]

2. For \( x = 10 \):
\[ f(10) = \max(4(10) + 3, 3(10) + 6) = \max(40 + 3, 30 + 6) = \max(43, 36) = 43 \]

### Step 5: Determine the maximum value of \( f(x) \)
Now we compare the values we calculated:
- \( f(-6) = -12 \)
- \( f(3) = 15 \)
- \( f(10) = 43 \)

The maximum value of \( f(x) \) in the interval \([-6, 10]\) is:
\[ \text{Maximum value} = 43 \]

### Final Answer
The maximum value of \( f(x) \) is \( 43 \).

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