How do you find all critical point and determine the min, max and inflection given #f(x)=x^3-6x^2+9x+8#?

Answer 1

See below

Given #f(x)#, his critical point are given by #f´(x)=0#. Lets calculate
#f´(x)=3x^2-12x+9=0#
Using quadratic formula #x=(12+-sqrt(144-108))/6=(12+-6)/6#
One critical point is #x=3# and the other is #x=1#
Analyzing the sign of #f´(x)# in intervals #(-oo,1)# #(1,3)# and #(3,+oo)# we found
#f´(x)>0# in #(-oo,1)# so f is increasing there #f´(x)<0# in #(1,3)# so f is decreasing there #f´(x)>0# in #(3,+oo)# so f is increasing there
Sumarizing #f(x)# has a maximum in #x=1#, has a minimum in #x=3# graph{x^3-6x^2+9x+8 [-15.25, 25.33, -3, 17.27]}
If we calculate #f´´(x)=6x-12=0# we found x=2 as infexion point
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Answer 2

To find critical points, we first find the derivative of the function and then solve for ( x ) when the derivative equals zero.

( f'(x) = 3x^2 - 12x + 9 )

To find the critical points, we solve for ( x ) when ( f'(x) = 0 ):

( 3x^2 - 12x + 9 = 0 )

( x^2 - 4x + 3 = 0 )

Factoring:

( (x - 3)(x - 1) = 0 )

( x = 3 ) or ( x = 1 )

To determine if these are minimum, maximum, or inflection points, we examine the sign of the second derivative, ( f''(x) ), at these critical points.

( f''(x) = 6x - 12 )

( f''(1) = 6(1) - 12 = -6 )

Since ( f''(1) ) is negative, the point ( x = 1 ) corresponds to a local maximum.

( f''(3) = 6(3) - 12 = 6 )

Since ( f''(3) ) is positive, the point ( x = 3 ) corresponds to a local minimum.

To determine if there are any inflection points, we look for changes in concavity. We set ( f''(x) = 0 ) and solve for ( x ):

( 6x - 12 = 0 )

( x = 2 )

We test the concavity of the function around ( x = 2 ) by evaluating ( f''(x) ) on either side of ( x = 2 ):

( f''(1) = -6 ) and ( f''(3) = 6 )

Since the sign changes from negative to positive, there is an inflection point at ( x = 2 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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