Question

select the correct answer.
what are the maximum and minimum of the function $f(x) = 0.9|-(x - 5)| + 7$?
a. maximum at $(5,7)$ and minimum at $(0,0)$
b. minimum at $(5,7)$ and no maximum
c. minimum at $(0,0)$ and no maximum
d. maximum at $(5,7)$ and no minimum

Explanation:

Step1: Simplify the function

The function is \( f(x) = 0.9|- (x - 5)| + 7 \). Since \( |-a|=|a| \), this simplifies to \( f(x)=0.9|x - 5|+7 \).

Step2: Analyze the absolute - value function

The general form of an absolute - value function is \( y = a|x - h|+k \), where \( (h,k) \) is the vertex. For the function \( y = |x - h| \), the absolute - value of a real number is always non - negative, i.e., \( |x - h|\geq0 \) for all real numbers \( x \).
In our function \( f(x)=0.9|x - 5|+7 \), the coefficient \( a = 0.9>0 \). When \( a>0 \), the graph of \( y=a|x - h|+k \) opens upwards. The vertex of the function \( y = a|x - h|+k \) is at the point \( (h,k) \). Here, \( h = 5 \) and \( k = 7 \), so the vertex is at \( (5,7) \).
Since the graph opens upwards, the function has a minimum value at the vertex. The minimum value of \( f(x) \) occurs when \( |x - 5| = 0 \) (because \( |x - 5|\geq0 \) and \( a = 0.9>0 \), so the smallest value of \( 0.9|x - 5| \) is \( 0 \), which occurs when \( x = 5 \)). When \( x = 5 \), \( f(5)=0.9\times0 + 7=7 \).
As \( x \) moves further away from \( x = 5 \) (either to the left or to the right), the value of \( |x - 5| \) increases, and since \( a=0.9>0 \), the value of \( 0.9|x - 5| \) also increases. So, \( f(x)=0.9|x - 5|+7 \) will keep increasing as \( |x - 5| \) increases, and there is no upper bound on how large \( f(x) \) can get. So, the function has a minimum at \( (5,7) \) and no maximum.

Answer:

B. minimum at (5,7) and no maximum