Sketch a graph of the following polynomial. Identify local extrema, inflection p

Question

Sketch a graph of the following polynomial. Identify local extrema, inflection points, and x- and y-intercepts when they exist. fox)-x3-6x2+9x Determine the local extrema. Select the correct choice and, if necessary, fill in the answer box(es) to complete your choice O A. The local minimummin ma isare at the port s There is no local maximum. O B. The local maximum/maxima is/are at the points). There is no local minimum. O c. The local maximum maxima is are at the point s and the local minimum minima i are at the point s O D. The function has no local extrema. What is(are) the inflection point(s)? Select the correct choice below and, if necessary, fill in the answer box to complete your choice OA. The inflection pointte) istare) O B. The function f does not have any inflection points. What is(are) the x-intercept(s)? Select the correct chcice below and, if necessary, fill in the answer box to complete your choice. 0 A. The x-intercept(s) is(are) the point(s) Type an ordered pair. Type an exact answer for each coordinate, using radicals as needed. Use a comma to separate answers as needed.) Type an ordered pair. Type an exact answer for each coordinate, using radicals as needed. Use a comma to separate answers as needed.) Type an ordered pair. Type an exact answer for each coordinate, using radicals as needed. Use a comma to separate answers as needed.) Type an ordered pair. Type an exact answer for each coordinate, using radicals as needed. Use a comma to separata answers as needed.) Type an ordered pair. Type an exact answer for each coordinate, using radicals as needed. Use a comma to separate answers as needed.) B. The function f does not have any x-intercepts What is the y-intercept? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A· The y-intercept is the point I Type an ordered pair. Type an exact answer for each coordinate, using radicals as needed. )

Explanation / Answer

x^3 - 6x^2 + 9x

Deriving :
f'= 3x^2 - 12x + 9 = 0
x^2 - 4x + 3 = 0
x = 1 and 3

Region1:(-inf , 1)
Test = 0
f' comes out to 9
Positive
So, increasing here

Region 2 : (1,3)
Test = 2
f'comes out -3, negative'
So, decreasing here

Region 3 : (3,inf)
Test = 4
Increasing

So, local max at x = 1
and local min at x = 3

When x = 1, y = 4
When x =3, y = 0

So, local max = (1,4)
local min = (3,0)

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Inflexion :
Deriving again :
f'= 3x^2 - 12x + 9
f'' = 6x - 12 = 0
x = 2

When x = 2, we get y = 2

So, inf pt is (2,2)

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x-int :

x^3 - 6x^2 + 9x = 0
x(x^2 - 6x + 9) = 0
x(x-3)(x-3) = 0
x = 0 and 3

(0,0) and (3,0)

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y-int :
Plug in x = 0, we get y = 0

(0,0)

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Graph :
Option A

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