1. Find the minimum and maximum values for the function with the given domain interval. [tex]\[ k(x) = 10^x, \quad \text{for} \quad -3 \leq x \leq 1 \][/tex]
A. Minimum value: [tex]\(-3\)[/tex]; Maximum value: [tex]\(1\)[/tex]
B. Minimum value: None; Maximum value: None
C. Minimum value: [tex]\(0.1\)[/tex]; Maximum value: [tex]\(1,000\)[/tex]
D. Minimum value: [tex]\(0.001\)[/tex]; Maximum value: [tex]\(10\)[/tex]
To find the minimum and maximum values for the function [tex]\( k(x) = 10^x \)[/tex] within the given domain interval [tex]\([-3, 1]\)[/tex], we need to evaluate the function at the endpoints of the interval.
1. Identify the endpoints of the interval: - The minimum value in the domain is [tex]\( x = -3 \)[/tex]. - The maximum value in the domain is [tex]\( x = 1 \)[/tex].
2. Evaluate the function at these points: - For [tex]\( x = -3 \)[/tex]: [tex]\[
k(-3) = 10^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001
\][/tex] - For [tex]\( x = 1 \)[/tex]: [tex]\[
k(1) = 10^1 = 10
\][/tex]
3. Determine the minimum and maximum values: - The function attains its minimum value at [tex]\( x = -3 \)[/tex], which is [tex]\[
k(-3) = 0.001
\][/tex] - The function attains its maximum value at [tex]\( x = 1 \)[/tex], which is [tex]\[
k(1) = 10
\][/tex]
Therefore, the minimum value of the function [tex]\( k(x) \)[/tex] over the given interval is [tex]\( 0.001 \)[/tex] and the maximum value is [tex]\( 10 \)[/tex].
Hence, the correct answer is: - Minimum value [tex]\( = 0.001 \)[/tex] - Maximum value [tex]\( = 10 \)[/tex]
So, the answer is: [tex]\[
\boxed{\text{minimum value } = 0.001 \text{; maximum value } = 10}
\][/tex]
