Questions: Find and classify the absolute extrema of the function (g(x)=x^2-121) on the domain (D=(-11,11)). If the function doesn't have an absolute extremum, write None for your answer.
Now we should determine the behavior of (g(x)) near the bounds of the domain. To do this, evaluate the corresponding one-sided limits.
[
lim x rightarrow 1^- g(x)=square
lim x rightarrow-11^+ g(x)=square
]
Transcript text: Find and classify the absolute extrema of the function $\mathrm{g}(\mathrm{x})=\mathrm{x}^{2}-121$ on the domain $\mathrm{D}=(-11,11)$. If the function doesn't have an absolute extremum, write None for your answer.
Now we should determine the behavior of $\mathrm{g}(\mathrm{x})$ near the bounds of the domain. To do this, evaluate the corresponding one-sided limits.
\[
\begin{array}{l}
\lim _{x \rightarrow 1^{-}} g(x)=\square \\
\lim _{x \rightarrow-11^{+}} g(x)=\square
\end{array}
\]
Solution
Solution Steps
Step 1: Finding Critical Points
To find the critical points of the function \( g(x) = x^2 - 121 \), we first compute its derivative:
\[
g'(x) = 2x
\]
Setting the derivative equal to zero gives:
\[
2x = 0 \implies x = 0
\]
Thus, the only critical point in the domain \( D = (-11, 11) \) is \( x = 0 \).
Step 2: Evaluating the Function at Critical Points
Next, we evaluate the function at the critical point:
\[
g(0) = 0^2 - 121 = -121
\]
This value represents a candidate for the absolute extremum.
Step 3: Evaluating One-Sided Limits
We now evaluate the one-sided limits at the boundaries of the domain:
\[
\lim_{x \to 1^{-}} g(x) = g(1) = 1^2 - 121 = -120
\]
\[
\lim_{x \to -11^{+}} g(x) = g(-11) = (-11)^2 - 121 = 0
\]
Step 4: Classifying Absolute Extrema
We compare the values obtained:
At the critical point \( x = 0 \), \( g(0) = -121 \).
The limit as \( x \) approaches \( 1 \) from the left is \( -120 \).
The limit as \( x \) approaches \( -11 \) from the right is \( 0 \).
The absolute maximum value is \( 0 \) at \( x = -11 \) and the absolute minimum value is \( -121 \) at \( x = 0 \).
Final Answer
The absolute extrema of the function \( g(x) \) on the domain \( D = (-11, 11) \) are:
Absolute minimum: \( \boxed{-121} \) at \( x = 0 \)
Absolute maximum: \( \boxed{0} \) at \( x = -11 \)