To solve this problem, we will first determine the values of the constants \( c \) and \( d \) by using the conditions provided in the question and then analyze the options given.
1. Given: The graph of \( y=g(x) \) passes through the point \((-11,0)\).
This means that:
[tex]\[
g(-11) = c \sqrt{-11 + d} = 0
\][/tex]
Since the only way for \( c \sqrt{-11 + d} \) to equal zero is if the term inside the square root equals zero, we set up the equation:
[tex]\[
-11 + d = 0 \implies d = 11
\][/tex]
2. Given: \( g(11) < 0 \).
With \( d = 11 \), substituting \( g(11) \) into the function:
[tex]\[
g(11) = c \sqrt{11 + 11} = c \sqrt{22}
\][/tex]
Since it is given that \( g(11) < 0 \), we have:
[tex]\[
c \sqrt{22} < 0
\][/tex]
The term \(\sqrt{22}\) is a positive number, which implies that for \( c \sqrt{22} \) to be negative, \( c \) must be negative. Hence,
[tex]\[
c < 0
\][/tex]
With \( c < 0 \) and \( d = 11 \), we analyze the provided options:
- Option A: \( c < d \)
[tex]\[
d = 11 \implies c < 11
\][/tex]
Since we already established that \( c < 0 \), it naturally follows that \( c < 11 \). Thus, option A is true.
- Option B: \( c > d \)
[tex]\[
d = 11 \implies c > 11
\][/tex]
This is not possible since we have \( c < 0 \). Hence, option B is false.
- Option C: \( g(0) = -11 \)
[tex]\[
g(0) = c \sqrt{0 + 11} = c \sqrt{11}
\][/tex]
For \( g(0) \) to be -11, the following equation must hold:
[tex]\[
c \sqrt{11} = -11 \implies c = -\frac{11}{\sqrt{11}} = -\sqrt{11}
\][/tex]
However, we established that \( c \) is simply less than 0, without specifying that it fulfills this exact value unless proven otherwise. Therefore, proving this exact relationship is outside the basic requirements, and thus option C is not necessarily true.
- Option D: \( g(0) = 11 \)
[tex]\[
g(0) = c \sqrt{0 + 11} = c \sqrt{11}
\][/tex]
If \( g(0) = 11 \),
[tex]\[
c \sqrt{11} = 11 \implies c = \frac{11}{\sqrt{11}} = \sqrt{11}
\][/tex]
Given \( c < 0 \), this is contrary to our established \( c < 0 \). Thus, option D is false.
Therefore, the only correct assertion is Option A: [tex]\( c < d \)[/tex].
