The sum of all local minimum values of the function f(x) = cases 1 - 2x, & x < -1 \ 13(7 + 2|x|), & -1 x 2 \ 1118(x - 4)(x - 5), & x > 2 cases is
- 16772
- 15772
- 17172
- 13172
Answer
(2) 15772
Solution
For x < -1: f(x) = 1 - 2x is decreasing and f(x) 3 as x -1-.
f(-1) = 13(7 + 2) = 3 f is continuous at x = -1.
For -1 x 0: f(x) = 13(7 - 2x), decreasing.
For 0 x 2: f(x) = 13(7 + 2x), increasing.
f keeps decreasing through x = -1 (no extremum there), and x = 0 is a point of local minimum:
f(0) = 73
At x = 2: f(2) = 113 and f(2+) = 1118(-2)(-3) = 113 (continuous).
For x > 2: f'(x) = 1118(2x - 9), negative for 2 < x < 92
f increases up to x = 2 and then decreases, so x = 2 is a point of local maximum.
f'(x) = 0 at x = 92 and f''(x) = 119 > 0 local minimum:
f ( 92 ) = 1118 ( 12 ) (- 12 ) = - 1172
Sum of local minimum values = 73 - 1172
= 168 - 1172 = 15772
Answer: (2)