To solve the problem, let's carefully go through the conditions provided and determine the constants [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for the function [tex]\(f(x) = a \sqrt{x + b}\)[/tex]. We are given two key pieces of information:
1. The function passes through the point [tex]\((7, 0)\)[/tex], which implies [tex]\(f(7) = 0\)[/tex].
2. The function value at [tex]\(x = 14\)[/tex] is negative, i.e., [tex]\(f(14) < 0\)[/tex].
Let's start with the first piece of information:
### Step 1: Determine [tex]\(b\)[/tex]
At the point [tex]\((7, 0)\)[/tex]:
[tex]\[ f(7) = 0 \][/tex]
Substitute [tex]\(x = 7\)[/tex] into the function:
[tex]\[ f(7) = a \sqrt{7 + b} = 0 \][/tex]
For this equation to hold, the expression inside the square root must equal zero (since [tex]\(a\)[/tex] cannot be zero):
[tex]\[ \sqrt{7 + b} = 0 \][/tex]
Squaring both sides, we get:
[tex]\[ 7 + b = 0 \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ b = -7 \][/tex]
### Step 2: Determine [tex]\(a\)[/tex]
Next, we use the second condition: [tex]\(f(14) < 0\)[/tex].
Substitute [tex]\(x = 14\)[/tex] and [tex]\(b = -7\)[/tex] into the function:
[tex]\[ f(14) = a \sqrt{14 + (-7)} < 0 \][/tex]
[tex]\[ f(14) = a \sqrt{7} < 0 \][/tex]
Since [tex]\(\sqrt{7}\)[/tex] is a positive number, for the product [tex]\(a \sqrt{7}\)[/tex] to be negative, [tex]\(a\)[/tex] must be negative.
### Conclusion
Given these analyses, we determined the following constants:
- [tex]\(b = -7\)[/tex]
- [tex]\(a\)[/tex] must be a negative value.
Thus, the constants [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy the given conditions are:
[tex]\[ a = -1 \][/tex]
[tex]\[ b = -7 \][/tex]
So, the required constants for the function [tex]\(f(x) = a \sqrt{x + b}\)[/tex] are:
[tex]\[ a = -1 \][/tex]
[tex]\[ b = -7 \][/tex]
These values ensure that the function passes through the point [tex]\((7, 0)\)[/tex] and satisfies [tex]\(f(14) < 0\)[/tex].
