Questions: Identify the function's local extreme values in the given domain, and say where they occur. Graph the function over the given domain. Which of the extreme values, if any, are absolute? f(x)=sqrt(100-x^2), -10 ≤ x ≤ 10 a. Find each local maximum, if any. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function has a local maximum value at one value of x. The maximum value is f(0)=10. (Type integers or simplified fractions.) B. The function has a local maximum value at three values of x. In increasing order of x-value, the maximum values are f()=, f()=, and f()=. (Type integers or simplified fractions.) C. The function has a local maximum value at two values of x. In increasing order of x-value, the maximum values are f()= and f()=. (Type integers or simplified fractions.) D. There are no local maxima. Find each local minimum, if any. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function has a local minimum value at one value of x. The minimum value is f()=. (Type integers or simplified fractions.) B. The function has a local minimum value at two values of x. In increasing order of x-value, the minimum values are f() and f()= . (Type integers or simplified fractions.) C. The function has a local minimum value at three values of x. In increasing order of x-value, the minimum values are f()= and f()=. (Type integers or simplified fractions.) D. There are no local minima.

Identify the function's local extreme values in the given domain, and say where they occur. Graph the function over the given domain. Which of the extreme values, if any, are absolute? f(x)=sqrt(100-x^2), -10 ≤ x ≤ 10 a. Find each local maximum, if any. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function has a local maximum value at one value of x. The maximum value is f(0)=10. (Type integers or simplified fractions.) B. The function has a local maximum value at three values of x. In increasing order of x-value, the maximum values are f()=, f()=, and f()=. (Type integers or simplified fractions.) C. The function has a local maximum value at two values of x. In increasing order of x-value, the maximum values are f()= and f()=. (Type integers or simplified fractions.) D. There are no local maxima. Find each local minimum, if any. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function has a local minimum value at one value of x. The minimum value is f()=. (Type integers or simplified fractions.) B. The function has a local minimum value at two values of x. In increasing order of x-value, the minimum values are f() and f()= . (Type integers or simplified fractions.) C. The function has a local minimum value at three values of x. In increasing order of x-value, the minimum values are f()= and f()=. (Type integers or simplified fractions.) D. There are no local minima.
Transcript text: Identify the function's local extreme values in the given domain, and say where they occur. Graph the function over the given domain. Which of the extreme values, if any, are absolute? \[ f(x)=\sqrt{100-x^{2}},-10 \leq x \leq 10 \] a. Find each local maximum, if any. Select the correct choice below and, if necessary, fill in the answe boxes to complete your choice. A. The function has a local maximum value at one value of $x$. The maximum value is $f(0)=10$. (Type integers or simplified fractions.) B. The function has a local maximum value at three values of $x$. In increasing order of $x$-value, the maximum values are $f(\quad)=\square, f(\quad)=\square$, and $f(\square)=\square$. (Type integers or simplified fractions.) C. The function has a local maximum value at two values of $x$. In increasing order of $x$-value, the maximum values are $f(\square)=\square$ and $f(\square)=\square$. (Type integers or simplified fractions.) D. There are no local maxima. Find each local minimum, if any. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function has a local minimum value at one value of $x$. The minimum value is $\mathrm{f}(\square)=\square$. (Type integers or simplified fractions.) B. The function has a local minimum value at two values of $x$. In increasing order of $x$-value, the minimum values are $f($ $\square$ and $f(\square)=$ $\square$ . (Type integers or simplified fractions.) C. The function has a local minimum value at three values of $x$. In increasing order of $x$-value, the minimum values are $\mathrm{f}($ $(\square)=$ and $f(\square)=$ (Type integers or simplified fractions.) D. There are no local minima.
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Solution

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Solution Steps

Step 1: Identify the function's local extreme values

The function given is \( f(x) = \sqrt{100 - x^2} \) with the domain \(-10 \leq x \leq 10\).

To find the local extrema, we first consider the derivative of the function: \[ f'(x) = \frac{-x}{\sqrt{100 - x^2}} \]

Setting the derivative equal to zero to find critical points: \[ \frac{-x}{\sqrt{100 - x^2}} = 0 \] This implies \( x = 0 \).

Evaluating the function at this critical point: \[ f(0) = \sqrt{100 - 0^2} = 10 \]

Since the function is symmetric and the derivative changes sign around \( x = 0 \), this is a local maximum.

Step 2: Determine the local maximum

The function has a local maximum value at one value of \( x \). The maximum value is \( f(0) = 10 \).

Step 3: Determine the local minimum

The function \( f(x) = \sqrt{100 - x^2} \) is non-negative and reaches its minimum value at the endpoints of the domain. Evaluating at the endpoints: \[ f(-10) = \sqrt{100 - (-10)^2} = 0 \] \[ f(10) = \sqrt{100 - 10^2} = 0 \]

Thus, the function has local minimum values at two values of \( x \). In increasing order of \( x \)-value, the minimum values are \( f(-10) = 0 \) and \( f(10) = 0 \).

Final Answer

  • Local maximum: \( f(0) = 10 \)
  • Local minima: \( f(-10) = 0 \) and \( f(10) = 0 \)

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