2.)The derivative f\'(x) is given. Use this information to find each critical nu
ID: 2888504 • Letter: 2
Question
2.)The derivative f'(x) is given. Use this information to find each critical number and classify each critical number of f(x) as a relative maximum, a relative minimum, or neither.
3.) The derivative f'(x) is given. Use this information to find each critical number and classify each critical number of f(x) as a relative maximum, a relative minimum, or neither.
4.)Match the graph of the function to the graph of its derivative.
5.)Match the graph of the function to the graph of its derivative
f (z) = (3-z) (z + 1)2Explanation / Answer
2)
I am assumin' that in question ,2
the derivative was given
So, f ' (x) = 0
x = 0 , x= 3 and x = -1
Region 1 : (-inf , -1)
Test = -2
The derivative itself is defined only for all x >= 0
So, only criticals are 0 and 3
Region 1 : (0 , 3)
Test = 1
We find f ' > 0
So, increasing here
Region 2 : (3, inf)
Test = 4
WE find f' < 0
So, decreasing here
So, at x = 3,
inc changed to dec
So, x = 3 is a relative max
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3)
Again, critical is a point where f' = 0 or undefined
Solving,
f' = 0, we get
x = 0 or x= 2
Region 1 : (-inf , 0)
Test= -1
WE find f' < 0
So, decreasing here
Region 2 : (0 , 2)
Test = 1
We find f' > 0
Increasing here
Region 3 : (2, inf)
Test = 3
We find f'>0
So, increasing here
Notice at x = 0, dec changed to inc
So x = 0 is a min
At x = 2, increasing remains increasing
So, x= 0 is what is called a saddle point
So, ans :
x = 0 and x = 2 are the criticals
x = 0 is a local min
x = 2 is a saddle
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