# Is #f(x)=-2x^5-3x^4+15x-4# concave or convex at #x=-4#?

Convex (sometimes called "concave upwards").

The concavity and convexity of a function can be determined by examining the sign of a function's second derivative.

Note that: you may call concave "concave down" and convex "concave up."

We must find the function's second derivative through the power rule:

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To determine the concavity of a function at a specific point, we need to examine the second derivative of the function at that point. If the second derivative is positive, the function is concave upward (convex) at that point. If the second derivative is negative, the function is concave downward at that point. If the second derivative is zero, the test is inconclusive.

Given ( f(x) = -2x^5 - 3x^4 + 15x - 4 ), we need to find the second derivative and evaluate it at ( x = -4 ).

First, find the first derivative of ( f(x) ): [ f'(x) = -10x^4 - 12x^3 + 15 ]

Now, find the second derivative: [ f''(x) = -40x^3 - 36x^2 ]

Evaluate the second derivative at ( x = -4 ): [ f''(-4) = -40(-4)^3 - 36(-4)^2 = -40(-64) - 36(16) = 2560 - 576 = 1984 ]

Since ( f''(-4) = 1984 ) which is positive, the function is concave upward (convex) at ( x = -4 ).

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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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