# Is #f(x)=-3x^5-2x^4-6x^3+x-7# concave or convex at #x=5#?

convex (concave DOWNWARD)

It's a pretty straightforward polynomial, so taking the derivative is pretty easy. You can just derive the terms one by one, proceeding left to right:

And then, do it again:

You can plug in x = 5 and calculate if you want, but if you're pressed for time on an exam it's not really necessary. Each term in the second derivative is negative for x=5, so the second derivative is therefore negative at that point, so it's convex (or concave downward) at x = 5.

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To determine the concavity or convexity of ( f(x) = -3x^5 - 2x^4 - 6x^3 + x - 7 ) at ( x = 5 ), you need to find the second derivative of the function and then evaluate it at ( x = 5 ).

First, find the first derivative: [ f'(x) = -15x^4 - 8x^3 - 18x^2 + 1 ]

Then, find the second derivative: [ f''(x) = -60x^3 - 24x^2 - 36x ]

Evaluate ( f''(5) ): [ f''(5) = -60(5)^3 - 24(5)^2 - 36(5) = -750 - 600 - 180 = -1530 ]

Since the second derivative is negative at ( x = 5 ), the function is concave down at ( x = 5 ).

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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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