absolute max and min

Question

Find the absolute maximum and absolute minimum values of the function  𝑓(𝑥)=|𝑥−1|+|𝑥+3| on the interval  −5≤𝑥≤5.

Explain your answer, and draw the graph of  𝑓.

Answer 1

If - 5 ≤ x < - 3 then f(x) = - x + 1 - x - 3 = - 2x - 2; (f'(x) = - 2 < 0 decreasing)

If - 3 ≤ x ≤ 1 then f(x) = - x + 2 + x + 3 = 5 (constant) abs min

If 1 < x ≤ 5 then f(x) = x - 1 + x + 3 = 2x + 2. If'(x) = 2 > 0 increasing)

So, f(-5) = 8

f(5) = 12 abs max