On which two intervals is the function increasing (enter intervals in ascending

Question

On which two intervals is the function increasing (enter intervals in ascending order)?
to ( , ) and  to ( , )
Find the region in which the function is positive: ( ) to ( )
Where does the function achieve its minimum? ( )

Please show your work. Thank you

For x E [-10, 12] the function f is defined by F (x) =x^5 (x+1)^4 On which two intervals is the function increasing (enter intervals in ascending order)? to ( , ) and to ( , ) Find the region in which the function is positive: ( ) to ( ) Where does the function achieve its minimum? ( ) Please show your work. Thank you

Explanation / Answer

f(x) = x^5 * (x + 1)^4

Deriving using product rule :

u = x^5 , v = (x + 1)^4
u' = 5x^4 , v' = 4(x + 1)^3

uv' + u'v becomes :

4x^5(x + 1)^3 + 5x^4(x + 1)^4

Equate to 0 :

4x^5(x + 1)^3 + 5x^4(x + 1)^4 = 0

x^4*(x+1)^3*(4x + 5(x + 1)) = 0

x^4 * (x + 1)^3 * (9x + 5) = 0

x = 0 , x = -1 and x = -5/9

This splits the number line into the following regions :

[-10 , -1) , (-1 , -5/9) , (-5/9 , 0) and (0 , 12)

Region 1 : [-10 , -1)
Testvalue = -2
f'(x) = x^4 * (x + 1)^3 * (9x + 5)
Plug in x = -2 :
f'(-2) = 16*-1*-13 --> positive
So, increasing over [-10 , -1)

Similarly, decreasing over (-1 , -5/9)

Increasing over (-5/9 , 0)

And increasing over (0 , infinity)

So, the increasing intervals are :

(-10 , -1) and (0 , infinity) -----> FIRST ANSWER

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Find the region in which the function is positive.

x^5*(x + 1)^4 > 0

x^5 = 0 --> x = 0
x + 1 =0 --> x = -1

Critical numbers are 0 and -1

So, this splits the number line into (-10 , -1) , (-1 , 0) and (0 , 12)

Region 1 : -10 to -1
Testvalue = -2
f(x) = x^5*(x + 1)^4
f(-1) = (-2)^5 * (-2 + 1)^4 ---> -32 * 1 --> -32 ---> negative

Region 2 : -1 to 0
Testvalue = -0.5
f(-0.5) = (-0.5)^5 * (-0.5 + 1)^4 --> -0.001953125 --> negative

Region 3 : 0 to 12 :
Testvalue = 1
f(1) = 1^5 * (1 + 1)^4 --> positive

So, the function is positive :

from 0 to 12 ---> ANSWER

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Minimum :

Critical numbers were found to be x = 0 , x = -1 and x = -5/9

Lets test these

f(x) = x^5 * (x + 1)^4
f(0) = 0
f(-1) = (-1)^5 * (-1 + 1)^4 = 0
f(-5/9) = (-5/9)^5 * (-5/9 + 1)^4 --> -0.0020649398333706
Now check the endpoints...
f(-10) = (-10)^5 * (-10 + 1)^4 --> -656100000
f(12) = (12)^5 * (12 + 1)^4 --> 7106890752

From the above values, the function achieves its minimum at x = -10
If they want the answer as a coordinate, then enter (-10 , -656100000)

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