Graph the function, then find the extreme values of the function on the interval and indicate where they occur.
-$ y=|x+2|+|x-4| $ on the interval $ -\infty

Question

Graph the function, then find the extreme values of the function on the interval and indicate where they occur.
-$ y=|x+2|+|x-4| $ on the interval $ -\infty< x <\infty $
A) Absolute minimum is: 5 on the interval $( - 3,3 ] \quad$ 
B) Absolute minimum is: 6 on the interval $( - 3,5 ]$ 
C) Absolute minimum is: 7 on the interval $( - 2,4 ]$ 
D) Absolute minimum is: 6 on the interval $( - 2,4 ]$

Quizplus · Accepted Answer

To find the extreme values of the function, we need to analyze the behavior of the function in each interval separated by the points where the absolute value expressions change their sign. These points are x = -2 and x = 4. When x < -2, both expressions inside the absolute value signs are negative, so we have: y = -(x+2) - (x-4) y = -2x - 2 When -2 < x < 4, the expression inside the first absolute value sign is negative, while the expression inside the second absolute value sign is positive, so we have: y = -(x+2) + (x-4) y = -2 When x > 4, both expressions inside the absolute value signs are positive, so we have: y = (x+2) + (x-4) y = 2x - 2 Thus, the extreme values occur at the endpoints of each interval: - For x < -2, the function decreases without bound as x approaches negative infinity, so there is no absolute minimum. - For -2 < x < 4, the absolute minimum occurs at x = -2 and is y = -2. - For x > 4, the function increases without bound as x approaches positive infinity, so there is no absolute maximum. Therefore, the only correct choice is D, where the absolute minimum is 6 and it occurs on the interval (-2,4].

Graph the Function, Then Find the Extreme Values of the Function