Questions: Find and classify the absolute extrema of the function g(x)=x^2-9 on the domain D=(-3,3). If the function doesn't have an absolute extremum, write None for your answer.

Find and classify the absolute extrema of the function g(x)=x^2-9 on the domain D=(-3,3). If the function doesn't have an absolute extremum, write None for your answer.

Transcript text: Find and classify the absolute extrema of the function $\mathrm{g}(\mathrm{x})=\mathrm{x}^{2}-9$ on the domain $\mathrm{D}=(-3,3)$. If the function doesn't have an absolute extremum, write None for your answer.

Solution

Solution Steps

Step 1: Find critical points

Find the derivative of g(x): g'(x) = 2x. Set g'(x) = 0 to find critical points: 2x = 0, so x = 0.

Step 2: Evaluate at critical points and endpoints

Since the interval is open, (-3, 3), we don't have endpoints to evaluate. Evaluate g(x) at the critical point x=0: g(0) = 0² - 9 = -9.

Step 3: Analyze the behavior near the endpoints

As x approaches -3 or 3, g(x) approaches (-3)² - 9 = 0 and 3² - 9 = 0, respectively. Since the interval is open, the function never actually reaches 0.

Final Answer

Absolute minimum value: -9 Absolute maximum value: None

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