Ex 1. Find the intervals where the function y = f(x) is increasing, decreasing, or is constant. y 4 X -6 -4 -2 2 4 -2 6

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A Increasing and Decreasing Functions
A function f is increasing over the interval
Ex 1. Find the intervals where the function y = f(x) is increasing,
decreasing, or is constant.
(a,b) if f(x1)< f(x2)whenever x, < x2 in the
interval (a,b).
y
7 (x2)
7 (x1 )
X2
A function f is decreasing over the interval
(a,b) if f(x1)> f (x2) whenever x <x2 in the
interval (a,b) .
X
-6
-4
-2
2
4
6.
-2
f(x1)
7(x2) ---
X2
A function f is constant over the interval
(a,b) if f(x1)= f(x2) for every x, and x, in the
interval (a,b).
|f (x)) = f (x2)
X2
B Test for Intervals of Increase or Decrease Ex 2. Find the intervals of increase or decrease for
Let y = f(x) be a differentiable function over
((x) = 2x³ + 3x² –12x.
(а,b). Then:
If f'(x)>0 for all xe (a,b) then f is increasing
over (a,b).
If f'(x)< 0 for all xe (a,b) then f is decreasing
over (a,b).
If f'(x)= 0 for all xe (a,b) then f is constant
over (a,b) .
Transcribed Image Text:A Increasing and Decreasing Functions A function f is increasing over the interval Ex 1. Find the intervals where the function y = f(x) is increasing, decreasing, or is constant. (a,b) if f(x1)< f(x2)whenever x, < x2 in the interval (a,b). y 7 (x2) 7 (x1 ) X2 A function f is decreasing over the interval (a,b) if f(x1)> f (x2) whenever x <x2 in the interval (a,b) . X -6 -4 -2 2 4 6. -2 f(x1) 7(x2) --- X2 A function f is constant over the interval (a,b) if f(x1)= f(x2) for every x, and x, in the interval (a,b). |f (x)) = f (x2) X2 B Test for Intervals of Increase or Decrease Ex 2. Find the intervals of increase or decrease for Let y = f(x) be a differentiable function over ((x) = 2x³ + 3x² –12x. (а,b). Then: If f'(x)>0 for all xe (a,b) then f is increasing over (a,b). If f'(x)< 0 for all xe (a,b) then f is decreasing over (a,b). If f'(x)= 0 for all xe (a,b) then f is constant over (a,b) .
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