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

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} ]

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} ]

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

Solve for ( x ).
The solutions will give you the xcoordinates of the inflection points.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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