Find the absolute extrema of the function (if any exist) on each interval. (If an answer does not exist, enter DNE.) \[ f(x)=\sqrt{25-x^{2}} \] (a) \( [-5,5] \) \[ \begin{array}{ll} \text { minimum (smaller } x \text {-value) } & (x, y)=(\square) \\ \text { minimum (larger } x \text {-value) } & (x, y)=(\square) \\ \text { maximum } & (x, y)=(\square) \end{array} \] (b) \( [-5,0) \) minimum \( (x, y)=(\square) \) maximum \( (x, y)=(\square) \) (c) \( (-5,5) \) minimum \( (x, y)=(\square) \) maximum \( (x, y)=(\square) \) (d) \( [1,5) \) minimum \( (x, y)=(\square) \) maximum \( (x, y)=(\square) \)

Let's evaluate the function \( f(x) = \sqrt{25 - x^2} \) for the given intervals. ### (a) For the interval \([-5, 5]\): Evaluate \( f \) at the endpoints: - \( f(-5) = \sqrt{25 - (-5)^2} = \sqrt{0} = 0 \) - \( f(5) = \sqrt{25 - 5^2} = \sqrt{0} = 0 \) Now check for critical points by taking the derivative: \[ f'(x) = \frac{-x}{\sqrt{25 - x^2}} \] Set \( f'(x) = 0 \): - This occurs at \( x = 0 \). - \( f(0) = \sqrt{25 - 0^2} = \sqrt{25} = 5 \). Thus, on the interval \([-5, 5]\): - Minimum (smaller \( x \)-value) \( (x, y) = (-5, 0) \) - Minimum (larger \( x \)-value) \( (x, y) = (5, 0) \) - Maximum \( (x, y) = (0, 5) \) ### (b) For the interval \([-5, 0)\): Endpoints evaluation: - \( f(-5) = 0 \) (from above) - As \( x \) approaches \( 0 \), \( f\) approaches \( 5 \) since \( f(0) \) is not included in the interval. Thus: - Minimum \( (x, y) = (-5, 0) \) - Maximum \( (x, y) = (0, 5) \) ### (c) For the interval \((-5, 5)\): No need to evaluate endpoints since they are not included: - Minimum occurs at \( x = -5 \): \( f(-5) = 0 \) - Maximum \( (x, y) = (0, 5) \) (as discussed). Thus: - Minimum \( (x, y) = (-5, 0) \) - Maximum \( (x, y) = (0, 5) \) ### (d) For the interval \([1, 5)\): Endpoints evaluation: - \( f(1) = \sqrt{25 - 1^2} = \sqrt{24} \approx 4.9 \) - As \( x \) approaches \( 5 \), \( f \) approaches \( 0 \) since \( f(5) \) is not included. Thus: - Minimum \( (x, y) = (5, 0) \) - Maximum \( (x, y) = (1, \sqrt{24}) \) To summarize: (a) \[ \begin{array}{ll} \text { minimum (smaller } x \text {-value) } & (x, y) = (-5, 0) \\ \text { minimum (larger } x \text {-value) } & (x, y) = (5, 0) \\ \text { maximum } & (x, y) = (0, 5) \end{array} \] (b) \[ \text{minimum } (x, y) = (-5, 0) \\ \text{maximum } (x, y) = (0, 5) \] (c) \[ \text{minimum } (x, y) = (-5, 0) \\ \text{maximum } (x, y) = (0, 5) \] (d) \[ \text{minimum } (x, y) = (5, 0) \\ \text{maximum } (x, y) = (1, \sqrt{24}) \]