# Where does the graph of #y=(5x^4)-(x^5)# have an inflection point?

An inflection point is a location on the graph where concavity changes. We'll examine concavity by examining the second derivative's sign:

A point on the graph is an inflection point, so we require:

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To find the inflection points of the function ( y = 5x^4 - x^5 ), we need to find where the second derivative changes sign or where the second derivative is zero.

First, let's find the first derivative ( y' ): [ y' = \frac{dy}{dx} = 20x^3 - 5x^4 ]

Next, let's find the second derivative ( y'' ): [ y'' = \frac{d^2y}{dx^2} = 60x^2 - 20x^3 ]

Now, set ( y'' ) to zero and solve for ( x ): [ 60x^2 - 20x^3 = 0 ] [ 20x^2(3 - x) = 0 ]

From this equation, ( x = 0 ) or ( x = 3 ).

Now, test the sign of ( y'' ) around these values to determine the inflection points:

For ( x < 0 ): Pick ( x = -1 ) (for instance): [ y'' = 60(-1)^2 - 20(-1)^3 = 60 - 20 = 40 ] Since ( y'' > 0 ), there's no inflection point to the left of ( x = 0 ).

For ( 0 < x < 3 ): Pick ( x = 1 ) (for instance): [ y'' = 60(1)^2 - 20(1)^3 = 60 - 20 = 40 ] Since ( y'' > 0 ), there's no inflection point between ( x = 0 ) and ( x = 3 ).

For ( x > 3 ): Pick ( x = 4 ) (for instance): [ y'' = 60(4)^2 - 20(4)^3 = 960 - 320 = 640 ] Since ( y'' > 0 ), there's no inflection point to the right of ( x = 3 ).

Therefore, the function ( y = 5x^4 - x^5 ) has an inflection point at ( x = 3 ).

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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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