Question

What is the minimum value of the function g(x) = |x – 4| + 8 on the interval [-2, 1]?

A. 14
B. 5
C. -2
D. 11

Answer 1

`|x-4|` is non-negative, and vanishes when
`x=4`.

For all
`x\in[-2,1]`, we have
`x<4`, so by definition of absolute value,

`g(x) = |x-4| + 8 = -(x-4)+8 = -x+4`

which is strictly decreasing on [-2, 1] since
`g'(x)=-1<0`. Then the minimum value of
`g` on [-2, 1] occurs at the endpoint
`x=1`, so that

`\min\left\{g(x) \mid -2\le x\le1\right\} = g(1) = |1-4| + 8 = \boxed{11}`