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

Answer 1

Critical Points are:
#(-1,1)# min
#(1/3,-5/27)# max

We have # f(x) = x^3 + x^2 - x #

To identify the critical vales, we differentiate and find find values of #x# st #f'(x)=0#

# { ( f'(x) < 0, => f(x) " is decreasing" ), ( f'(x) = 0, => f(x) " is stationary" ), ( f'(x) > 0, => f(x) " is increasing" ) :} #

Differentiating wrt #x#' we have:

# f'(x) = 3x^2 + 2x - 1 # .... [1]

At a critical point, # f'(x)=0 #

# f'(x)=0 => 3x^2 + 2x - 1 = 0 #
# :. (3x-1)(x+1) = 0 #
# x=-1,1/3 #

Ton find the y-coordinate we substitute the required value into #f(x)#
# x=-1 => f(-1)=-1+1-(-1)=1 #
# x=1/3 => f(2/3)=1/27+1/9+1/3=-5/27 #

So the critical points are #(-1,1)# and #(1/3,-5/27)#

We identify the nature of these critical points by looking at the sign of second derivative, and

# { ( f''(x) < 0, => f'(x) " is decreasing" => "maximum" ), ( f''(x) = 0, => f'(x) " is stationary" => "inflection" ), ( f''(x) > 0, => f'(x) " is increasing" => "minimum" ) :} #

Differentiating [1] wrt #x# gives s;

# f''(x) = 6x + 2 #
# x=-1 => f''(-2)=-6+2 < 0 #, ie a maximum
# x=1/3 => f''(1)= 2+2>0#, ie a minimum

Incidental, As this is a cubic with a positive coefficient of #x^3#, we can deduce that the critical point corresponding to the smallest value of #x# must be the maximum and that corresponding to the larger value must be the maximum. It is not possible to have any other possibility!

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Answer 2

The max is at #(-1,1)#
The min is at #(1/3,-5/27)#
The inflexion point is at #(-1/3,11/27)#

We have to calculate the first and second derivative.

#f(x)=x^3+x^2-x#
#f'(x)=3x^2+2x-1=(3x-1)(x+1)#
#f'(x)=0# when #x=1/3# and #x=-1#

So, we do a sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-1##color(white)(aaaa)##1/3##color(white)(aaaa)##+oo#
#color(white)(aaaa)##f'(x)##color(white)(aaaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaaa)##uarr##color(white)(aaaa)##darr##color(white)(aaaa)##uarr#
So, we have a max at #x=-1# and a min at #x=1/3#
To determine the inflexion points, we calculate #f''(x)#
#f''(x)=6x+2#
#f''(x)=0# when #x=-1/3#
The inflexion point is at #x=-1/3#
Also, #f''(-1)=-4<0# which is a max
and #f''(1/3)=4>0# which is a min

graph{x^3+x^2-x [-2.43, 2.436, -1.217, 1.215]}

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Answer 3

To find all critical points of ( f(x) = x^3 + x^2 - x ) and determine whether they correspond to a minimum, maximum, or point of inflection, follow these steps:

  1. Find the first derivative of ( f(x) ) and set it equal to zero to find critical points.
  2. Use the second derivative test to determine whether each critical point corresponds to a minimum, maximum, or point of inflection.

Let's proceed with the calculations:

  1. Find the first derivative: [ f'(x) = \frac{{d}}{{dx}}(x^3 + x^2 - x) ]

  2. Set ( f'(x) ) equal to zero and solve for ( x ) to find critical points.

  3. After finding critical points, find the second derivative: [ f''(x) = \frac{{d^2}}{{dx^2}}(x^3 + x^2 - x) ]

  4. Use the second derivative test:

    • If ( f''(x) > 0 ), the critical point corresponds to a local minimum.
    • If ( f''(x) < 0 ), the critical point corresponds to a local maximum.
    • If ( f''(x) = 0 ), the test is inconclusive, and further analysis is needed.

Let's solve for critical points and apply the second derivative test to determine the nature of each critical point.

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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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