# How do I find concavity and points of inflection for #y = 3x^5 - 5x^3#?

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Given -

Find the first derivative -

Then -

Find the second derivative -

The value of the function -

The value of the function -

The value of the function -

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To find the concavity and points of inflection for ( y = 3x^5 - 5x^3 ), follow these steps:

- Find the second derivative of the function ( y ).
- Set the second derivative equal to zero and solve for ( x ) to find any possible points of inflection.
- Determine the sign of the second derivative on intervals between these points to identify the concavity of the function.

Let's proceed with these steps:

- First derivative: ( y' = 15x^4 - 15x^2 )
- Second derivative: ( y'' = 60x^3 - 30x )
- Set ( y'' ) equal to zero and solve for ( x ): ( 60x^3 - 30x = 0 ) Factor out ( 30x ): ( 30x(x^2 - 1) = 0 ) Solve for ( x ): ( x = 0 ) (critical point), ( x = -1 ), ( x = 1 )
- Determine the sign of ( y'' ) in each interval:
- Interval ( (-\infty, -1) ): Pick ( x = -2 ) (test point), ( y''(-2) = -120 < 0 ) so concave down.
- Interval ( (-1, 0) ): Pick ( x = -0.5 ) (test point), ( y''(-0.5) = 15 > 0 ) so concave up.
- Interval ( (0, 1) ): Pick ( x = 0.5 ) (test point), ( y''(0.5) = 15 > 0 ) so concave up.
- Interval ( (1, \infty) ): Pick ( x = 2 ) (test point), ( y''(2) = 120 > 0 ) so concave up.

Points of inflection: ( x = -1 ) and ( x = 1 ) Concave up: ( (-1, 0) ) and ( (0, 1) ) Concave down: ( (-\infty, -1) ) and ( (1, \infty) )

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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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