# How do you find the x and y coordinates of all inflection points #f(x) = x^4 - 12x^2#?

The coordinates of the two inflection points are

This is the graph; try to locate the inflection points on it:

graph{[-10, 10, -40, 40]} x^4–x^2

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To find the inflection points of the function ( f(x) = x^4 - 12x^2 ), you first need to find the second derivative of the function and then solve for the points where the second derivative changes sign.

First derivative: [ f'(x) = 4x^3 - 24x ]

Second derivative: [ f''(x) = 12x^2 - 24 ]

Set the second derivative equal to zero and solve for ( x ): [ 12x^2 - 24 = 0 ] [ 12x^2 = 24 ] [ x^2 = 2 ] [ x = \pm \sqrt{2} ]

Now, test the intervals between these points and beyond them to determine the sign of the second derivative in each interval. You can use the second derivative test or simply plug in values from each interval into the second derivative.

For ( x < -\sqrt{2} ): Choose ( x = -3 ) [ f''(-3) = 12(-3)^2 - 24 = 12(9) - 24 = 108 - 24 = 84 > 0 ]

For ( -\sqrt{2} < x < \sqrt{2} ): Choose ( x = 0 ) [ f''(0) = 12(0)^2 - 24 = -24 < 0 ]

For ( x > \sqrt{2} ): Choose ( x = 3 ) [ f''(3) = 12(3)^2 - 24 = 12(9) - 24 = 108 - 24 = 84 > 0 ]

So, the function ( f(x) = x^4 - 12x^2 ) has inflection points at ( x = -\sqrt{2} ) and ( x = \sqrt{2} ). To find the corresponding ( y ) coordinates, plug these values into the original function:

For ( x = -\sqrt{2} ): [ f(-\sqrt{2}) = (-\sqrt{2})^4 - 12(-\sqrt{2})^2 = 2 - 12(2) = -22 ]

For ( x = \sqrt{2} ): [ f(\sqrt{2}) = (\sqrt{2})^4 - 12(\sqrt{2})^2 = 2 - 12(2) = -22 ]

So, the inflection points are ( (-\sqrt{2}, -22) ) and ( (\sqrt{2}, -22) ).

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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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