Definitions
Connecting f(x) and f' (x)
Graph of f'(x)
Function Analysis Algebraically
100
If a functions FIRST derivative is negative on a given interval, what does that tell you about the function?
The function is DECREASING on the interval.
100
For a function f(x), f'(-3) = 5 indicates that f(x) is __________ at x = -3.
Increasing
100
The graph of f' the derivative of f is shown below. Determine all intervals on which f is increasing on the closed interval [-3, 4].
(-3, -2)
100
Find the critical values of f(x). f(x) = 2x4 -4x2+1.
x = -1, 0, 1
200
If a functions FIRST derivative is positive on a given interval, what does that tell you about the function?
The function is INCREASING on the interval.
200
For a function g(x), g' (3) = -8 indicates that g(x) is ______________ at x = 3.
Decreasing
200
The graph of f' the derivative of f is shown below. Determine all the intervals where f is decreasing on the closed interval [-3, 4],
(-2, 1) U (1, 4)
200
Find all the intervals on which f(x) is increasing.
f(x) = 2x3+6x2-29
(-00,-2)U(0, 00)
300
How do you determine critical values of a function?
When f' (x) =0, f' (x) DNE, or end points.
300
Use the sign chart to determine where the function is increasing.
(-00, -2)U(4, 00)
300
The graph of f' the derivative of f is shown below. Determine all critical values of f on the closed interval [-3, 4].
x= -3, -2, 1, 4
300
Find all the intervals on which f(x) is decreasing.
f(x) = 5x4-4x3-23
(-00, 0 )U(0,3/5)
400
When does a relative maximum occur at x = a?
When f' (x) changes from positive to negative.
400
Use the sign chart to determine where the function is decreasing.
(-2, 4)
400
The graph of f' the derivative of f is shown below. Determine the values of x, if any, at which the function f has a relative minimum.
no minimum
400
Find the x-coordinates of all relative minima of f(x).
f(x) = 2x4+24x3+25
x = -9
500
When does a relative minimum occur at x = a?
When f'(x) changes from negative to positive.
500
Use the sign chart to determine where the function has a relative minimum.
At x=4
500
The graph of f' the derivative of f is shown below. Determine the values of x, if any, at which f has a relative maximum.
x = -2
500
Find the x-coordinates of all relative maxima of f(x).
f(x) = -2x4-16x3+23
x = -6