Download Finding Increasing & Decreasing Intervals f’(c) Steps * Find the 1

Finding Increasing & Decreasing Intervals
When f’(c) = 0 or is undefined → c is a critical number
Steps * Find the 1st derivative and set it equal to 0
* Solve for x to find the critical numbers
* Use test points (tp) to determine the sign of f’(x)
Example: f(x) = x2
If f’(x) < 0 then function is decreasing
2
X -9
on the interval (a,b)
If f’(x) > 0 then function is increasing
f’(x) = -18x = 0
on the interval (a,b)
2
2
If f’(x) = 0 then function is constant
(x - 9)
on the interval (a,b)
-18x = 0
(x2 – 9)2 = 0
x = 0 → critical number x = -3, 3 → discontinuities
-3
3
0
-10
-1
1
10
tp
tp
tp
tp
Use test points to find increasing and decreasing intervals
f’(-10) = + → increasing
f’(1) = − → decreasing
f’(-1) = + → increasing
f’(10) = − → decreasing
Therefore: Increasing on (-∞, -3), (-3, 0)
Decreasing on (0, 3), (3, ∞)
Relative max: (0,0)
Maximums and Minimums
Max @ (a, f(a)) if:
then x = a is a relative
maximum
(+) a (-)
Min @ (b, f(b)) if:
(+)
(-)
b
(-)
(+)
then x = b is a relative
minimum