Consider the functions:
[tex]
\begin{array}{l}
f(x)=\sqrt{x} \
g(x)=\sqrt{x-3}+1 \
h(x)=\sqrt{x+1}-2
\end{array}
[/tex]
Which statement compares the relative locations of the minimums of the functions?
A. The minimums of [tex]g(x)[/tex] and [tex]h(x)[/tex] are both in the first quadrant.
B. The minimums of [tex]g(x)[/tex] and [tex]h(x)[/tex] are both in the third quadrant.
C. The minimum of [tex]h(x)[/tex] is farther right and up from the minimums of [tex]f(x)[/tex] and [tex]g(x)[/tex].
D. The minimum of [tex]h(x)[/tex] is farther left and down from the minimums of [tex]f(x)[/tex] and [tex]g(x)[/tex].
Answer (2)
Find the minimum of f ( x ) = x at ( 0 , 0 ) .
Find the minimum of g ( x ) = x − 3 + 1 at ( 3 , 1 ) .
Find the minimum of h ( x ) = x + 1 − 2 at ( − 1 , − 2 ) .
Compare the locations of the minimums and conclude that the minimum of h ( x ) is farther left and down from the minimums of f ( x ) and g ( x ) .
The minimum of h ( x ) is farther left and down from the minimums of f ( x ) and g ( x ) .
Explanation
Understanding the Problem We are given three functions: f ( x ) = x , g ( x ) = x − 3 + 1 , and h ( x ) = x + 1 − 2 . We need to determine which statement correctly compares the locations of the minimums of these functions.
Finding the Minimum of f(x) The minimum value of f ( x ) = x occurs when x is at its smallest possible value. Since the square root function is only defined for non-negative values, the smallest value of x is 0. Thus, the minimum of f ( x ) is at ( 0 , 0 ) = ( 0 , 0 ) .
Finding the Minimum of g(x) The minimum value of g ( x ) = x − 3 + 1 occurs when x − 3 is at its smallest possible value. Again, since the square root function is only defined for non-negative values, the smallest value of x − 3 is 0, which means x = 3 . Thus, the minimum of g ( x ) is at ( 3 , 3 − 3 + 1 ) = ( 3 , 1 ) .
Finding the Minimum of h(x) The minimum value of h ( x ) = x + 1 − 2 occurs when x + 1 is at its smallest possible value. The smallest value of x + 1 is 0, which means x = − 1 . Thus, the minimum of h ( x ) is at ( − 1 , − 1 + 1 − 2 ) = ( − 1 , − 2 ) .
Analyzing the Locations of the Minimums Now we analyze the locations of the minimums. The minimum of f ( x ) is at ( 0 , 0 ) , which is the origin. The minimum of g ( x ) is at ( 3 , 1 ) , which is in the first quadrant. The minimum of h ( x ) is at ( − 1 , − 2 ) , which is in the third quadrant.
Comparing the Locations of the Minimums Let's evaluate the given statements:
The minimums of g ( x ) and h ( x ) are both in the first quadrant. This is false, since the minimum of h ( x ) is in the third quadrant.
The minimums of g ( x ) and h ( x ) are both in the third quadrant. This is false, since the minimum of g ( x ) is in the first quadrant.
The minimum of h ( x ) is farther right and up from the minimums of f ( x ) and g ( x ) . This is false. Comparing h ( x ) at ( − 1 , − 2 ) to f ( x ) at ( 0 , 0 ) , h ( x ) is to the left and down. Comparing h ( x ) at ( − 1 , − 2 ) to g ( x ) at ( 3 , 1 ) , h ( x ) is to the left and down.
The minimum of h ( x ) is farther left and down from the minimums of f ( x ) and g ( x ) . This is true.
Conclusion Therefore, the correct statement is: The minimum of h ( x ) is farther left and down from the minimums of f ( x ) and g ( x ) .
Examples
Understanding the minimum values of functions is crucial in various real-world applications. For instance, in business, minimizing cost functions helps companies determine the most efficient production levels. In physics, finding the minimum potential energy helps predict the stable states of a system. In engineering, minimizing stress functions ensures structural integrity. By analyzing functions and their minimums, we can optimize processes and make informed decisions in diverse fields.
The minimum of the functions f ( x ) , g ( x ) , and h ( x ) are located at ( 0 , 0 ) , ( 3 , 1 ) , and ( − 1 , − 2 ) respectively. The correct statement is that the minimum of h ( x ) is farther left and down from the minimums of f ( x ) and g ( x ) . Therefore, the answer is option D.
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