What are the points of inflection of #f(x)=x^3-3x^2-x+7 #?

Answer 1

At #x=1#

The point of inflection is the point where the second derivative of the function #f(x)# is zero i.e. #(d^2f)/(dx^2)=0#. At these points the slope of the curve changes from increasing to decreasing and vice versa.
Here function is #f(x)=x^3-3x^2-x+7#
and #(df)/(dx)=3x^2-6x-1# and #(d^2f)/(dx^2)=6x-6#
and #(d^2f)/(dx^2)# is zero, when #6x-6-0# or #x=1#
and at this #y=4#

graph{-5.043, 4.957, 1.54, 6.54]} = (x^3-3x^2-x+7-y)((x-1)^2+(y-4)^2-0.01)=0

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Answer 2

To find the points of inflection of the function ( f(x) = x^3 - 3x^2 - x + 7 ), follow these steps:

  1. Find the second derivative of the function ( f(x) ).
  2. Set the second derivative equal to zero and solve for ( x ) to find any possible points of inflection.
  3. Use the second derivative test or analyze the behavior of the function around these points to determine if they are points of inflection.

Let's proceed with these steps:

  1. Find the second derivative: [ f(x) = x^3 - 3x^2 - x + 7 ] [ f'(x) = 3x^2 - 6x - 1 ] [ f''(x) = 6x - 6 ]

  2. Set the second derivative equal to zero and solve for ( x ): [ 6x - 6 = 0 ] [ 6x = 6 ] [ x = 1 ]

  3. Determine if ( x = 1 ) is a point of inflection:

    • To do this, analyze the behavior of the function around ( x = 1 ).
    • Before and after ( x = 1 ), examine whether the concavity of the function changes.
    • You can also use the second derivative test: If ( f''(x) ) changes sign from positive to negative or vice versa at ( x = 1 ), then ( x = 1 ) is a point of inflection.

    We can see that ( f''(0) = -6 ) and ( f''(2) = 6 ), indicating a change in concavity at ( x = 1 ).

Therefore, the point of inflection of the function ( f(x) = x^3 - 3x^2 - x + 7 ) is ( x = 1 ).

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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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