A. Determine where each of the following functions is increasing and where it is decreasing.
1) f(x) = x^2 - 6x + 19
f’(x) = 2x – 6 > 0 for x > 3
Function increases on x > 3 and decreases on x < 3
2) f(x) = 10x - x^2
f’(x) = 10 – 2x > 0 for x < 5
Function increases on x < 5 and decreases on x > 5
B. Determine the critical values of each of the following functions:
1) f(x) = x^2 - 16x
f’(x) = 2x - 16 = 0 at x = 8
Critical value is f(8) = 8^2 - 16*8 = 64 – 128 = -64
Crit value: -64
2) f(x) = x^3 – 2
f’(x) = 3x^2 = 0 at x = 0
Critical value is f(0) = 0 - 2 = -2
Crit value: -2
C. Find all relative extreme points of each of the following functions:
1) f(x) = x^2 - 20x
f’(x) = 2x – 20 = 0 at x = 10
f(10) = 10^2 – 20*10 = 100 – 200 = -100
Minimum point is at (10, -100)
2) f(x) = x^3 - 3x – 2
f’(x) = 3x^2 – 3 = 0 at x = 1 and at x = -1
f(1) = 1 – 3 – 2 = -4, f(-1) = -1 + 3 – 2 = 0
Minimum point is at (1, -4) Maximum point is at (-1, 0)
3) f(x) = -x^3 - 3x^2 + 7
f’(x) = -3x^2 – 6x = 0 at x = 0 and at x = -2
f(0) = 0 – 0 + 7 = 7, f(-2) = -(-8) – 3(4) + 7 = 3
Minimum point is at (-2, 3) Maximum point is at (0, 7)
4) f(x) = x^4 - 2x^2 + 3
f’(x) = 4x^3 – 4x = 0 at x = 0 and at x = 1 and at x = -1
f(0) = 0 – 0 + 3 = 3, f(1) = (1) – 2(1) + 3 = 2, f(-1) = (1) – 2(1) + 3 = 2
Minimum points are at (-1, 2) and (1, 2) Maximum point is at (0, 3)