# What is the end behavior and turning points of #y = -2x^3 + 3x - 1#?

As

Turning points are

For end behaviour, note the leading coefficient and the degree. The degree is odd and the leading coefficient negative, hence it would rise to the left and fall to the right.

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To find the turning points, differentiate the function and set it to zero. This yields :

Putting all together, we get the graph :

graph{-2x^3+3x-1 [-10, 10, -5, 5]}

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The end behavior of the polynomial function ( y = -2x^3 + 3x - 1 ) is determined by the leading term, which is ( -2x^3 ). As ( x ) approaches positive or negative infinity, the leading term ( -2x^3 ) dominates the behavior of the function. Since the coefficient of the leading term is negative, the end behavior is as follows:

- As ( x ) approaches positive infinity, ( y ) approaches negative infinity.
- As ( x ) approaches negative infinity, ( y ) approaches negative infinity as well.

The turning points of the function correspond to the points where the derivative of the function ( y' ) equals zero. To find the turning points, we first find the derivative of ( y ) with respect to ( x ), and then solve for ( x ) when ( y' = 0 ). After finding these ( x )-values, we can evaluate ( y ) at these points to find the corresponding ( y )-values.

[ y' = \frac{d}{dx}(-2x^3 + 3x - 1) = -6x^2 + 3 ]

Setting ( y' = 0 ) to find critical points:

[ -6x^2 + 3 = 0 ] [ -6x^2 = -3 ] [ x^2 = \frac{1}{2} ] [ x = \pm \sqrt{\frac{1}{2}} = \pm \frac{\sqrt{2}}{2} ]

These critical points correspond to the potential turning points of the function. To find the corresponding ( y )-values, we substitute these ( x )-values back into the original function:

[ y\left(\frac{\sqrt{2}}{2}\right) = -2\left(\frac{\sqrt{2}}{2}\right)^3 + 3\left(\frac{\sqrt{2}}{2}\right) - 1 ] [ y\left(\frac{\sqrt{2}}{2}\right) = -2\left(\frac{1}{2\sqrt{2}}\right) + \frac{3\sqrt{2}}{2} - 1 ] [ y\left(\frac{\sqrt{2}}{2}\right) = -\frac{1}{\sqrt{2}} + \frac{3\sqrt{2}}{2} - 1 ] [ y\left(\frac{\sqrt{2}}{2}\right) = \frac{3\sqrt{2}}{2} - \frac{1}{\sqrt{2}} - 1 ] [ y\left(\frac{\sqrt{2}}{2}\right) = \frac{3\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - 1 ] [ y\left(\frac{\sqrt{2}}{2}\right) = \frac{2\sqrt{2}}{2} - 1 ] [ y\left(\frac{\sqrt{2}}{2}\right) = \sqrt{2} - 1 ]

Similarly,

[ y\left(-\frac{\sqrt{2}}{2}\right) = -\sqrt{2} - 1 ]

Therefore, the turning points are approximately ( \left(\frac{\sqrt{2}}{2}, \sqrt{2} - 1\right) ) and ( \left(-\frac{\sqrt{2}}{2}, -\sqrt{2} - 1\right) ).

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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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