Question
Identify the point at which the maximum of the square root function \( y=-\sqrt{3 x+6}-1 \) occurs. (1 point) \( (-6,-1) \) \( (-2,1) \) \( (-2,-1) \) \( (-6,1) \)
Ask by Rogers Collins. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The maximum of the function occurs at \( (-2, -1) \).
Solution
The maximum of the function \( y = -\sqrt{3x + 6} - 1 \) occurs where the expression inside the square root is minimized.
1. **Domain**: \( 3x + 6 \geq 0 \Rightarrow x \geq -2 \).
2. **Minimum of the square root**: The smallest value of \( \sqrt{3x + 6} \) is 0, which occurs at \( x = -2 \).
3. **Maximum of \( y \)**: Substituting \( x = -2 \) into the function gives:
\[
y = -\sqrt{3(-2) + 6} - 1 = -\sqrt{0} - 1 = -1
\]
So, the point where the maximum occurs is \( \boxed{(-2,\ -1)} \).
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Simplify this solution Extra Insights
To determine the point at which the maximum of the square root function \( y=-\sqrt{3 x+6}-1 \) occurs, we first identify the characteristics of the function. The expression inside the square root, \( 3x + 6 \), must be non-negative for the square root to be defined, which gives us \( 3x + 6 \geq 0 \) or \( x \geq -2 \). Next, since the square root function is being negated and shifted downwards by 1, the maximum value occurs at the smallest value of \( x \) within its domain, which is \( x = -2 \). Plugging this into the function: \[ y = -\sqrt{3(-2) + 6} - 1 = -\sqrt{0} - 1 = -1. \] Thus, the maximum point occurs at \( (-2, -1) \). The answer is \( (-2, -1) \).
