How do you find the inflection points for the function #f(x)=x/(x^2+9)#?

Answer 1
Inflection points on #y=f(x)# are places on the curve where the concavity (measured by #f''(x)#) changes from positive to negative, or vice versa. We can see where the second derivative is zero to look for possibilities. Let's take the first two derivatives:
#f(x) = x/(x^2+9)# now use the quotient rule (derivs in [ ])
#f'(x) = ((x^2+9)[1]-(x)[2x])/((x^2+9)^2)=(-x^2+9)/(x^2+9)^2#; do again:
#f''(x) = ((x^2+9)^2*[-2x]-(-x^2+9)[2(x^2+9)(2x)])/((x^2+9)^4)# # = ((x^2+9)*[-2x]-(-x^2+9)[2(2x)])/((x^2+9)^3)=(-54x+2x^3)/(x^2+9)^3#
If we set this equal to zero, and solve: #-54x+2x^3=0 => 2x(x^2-27)=0#; we get three answers: #x = 0 and x = +-sqrt(27) = +-3sqrt(3)#. The sign of #f''(x)# does change across #x = 0#, so the inflection points are at #x=0, +-sqrt(3)#.

To get the y-coordinates, enter these x values into the original function.

Moving forward: Sketch the graph now. You may also want to identify the critical points. I'll let you do that so your mind will work even harder!

\dansmath to the rescue!/

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Answer 2

To find the inflection points of the function ( f(x) = \frac{x}{x^2 + 9} ), you first need to find the second derivative of the function, then solve for the points where the second derivative equals zero or is undefined.

  1. Find the first derivative ( f'(x) ). [ f'(x) = \frac{d}{dx}\left(\frac{x}{x^2 + 9}\right) ] [ = \frac{(x^2 + 9) - x(2x)}{(x^2 + 9)^2} ] [ = \frac{9 - x^2}{(x^2 + 9)^2} ]

  2. Find the second derivative ( f''(x) ). [ f''(x) = \frac{d}{dx}\left(\frac{9 - x^2}{(x^2 + 9)^2}\right) ] [ = \frac{(-2x)(x^2 + 9)^2 - (9 - x^2)(2(x^2 + 9)(2x))}{(x^2 + 9)^4} ] [ = \frac{-2x(x^2 + 9)^2 - 4x(x^2 + 9)(9 - x^2)}{(x^2 + 9)^4} ]

  3. Set ( f''(x) ) equal to zero and solve for ( x ). [ -2x(x^2 + 9)^2 - 4x(x^2 + 9)(9 - x^2) = 0 ]

  4. Solve for ( x ).

The solutions will give you the x-coordinates of the inflection points.

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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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