How do you find the inflection points for the function #f(x)=x^3+x#?

Answer 1
#f(x)=x^3+x#

By taking derivatives,

#f'(x)=3x^2+1#
#f''(x)=6x=0 Rightarrow x=0#,
which is the #x#-coordinate of a possible inflection point. (We still need to verify that #f# changes its concavity there.)
Use #x=0# to split #(-infty,\infty)# into #(-infty,0)# and #(0,infty)#.
Let us check the signs of #f''# at sample points #x=-1# and #x=1# for the intervals, respectively. (You may use any number on those intervals as sample points.)
#f''(-1)=-6<0 Rightarrow f# is concave downward on #(-infty,0)#
#f''(1)=6>0 Rightarrow f# is concave upward on #(0,infty)#
Since the above indicates that #f# changes its concavity at #x=0#, #(0,f(0))=(0,0)# is an inflection point of #f#.

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Answer 2

To find the inflection points for the function (f(x) = x^3 + x), we need to find the points where the concavity changes. Inflection points occur where the second derivative changes sign or where the second derivative is zero.

  1. Find the first derivative of (f(x)): [f'(x) = 3x^2 + 1]

  2. Find the second derivative of (f(x)): [f''(x) = 6x]

  3. Set (f''(x) = 0) to find possible inflection points: [6x = 0] [x = 0]

  4. Test the sign of (f''(x)) around (x = 0):

    • For (x < 0), (f''(x) < 0), so the concavity is downward.
    • For (x > 0), (f''(x) > 0), so the concavity is upward.

Therefore, the function (f(x) = x^3 + x) has an inflection point at (x = 0).

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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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