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

convex at x = -2

To test if a function is concave / convex at f(a), require to find the value of f''(a).

• If f''(a) > 0 then f(x) is convex at x = a

• If f''(a) < 0 then f(x) is concave at x = a

= 160 - 48 + 24 = 136

since f''(-2) > 0 then f(x) is convex at x = -2 graph{(-x^5-x^4-2x^3+x-7) [-16.02, 16.02, -8.01, 8.01]}

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To determine whether the function ( f(x) = -x^5 - x^4 - 2x^3 + x - 7 ) is concave or convex at ( x = -2 ), we need to evaluate the second derivative of the function at that point.

First, find the first derivative of ( f(x) ): [ f'(x) = -5x^4 - 4x^3 - 6x^2 + 1 ]

Now, find the second derivative of ( f(x) ): [ f''(x) = -20x^3 - 12x^2 - 12x ]

Evaluate ( f''(-2) ): [ f''(-2) = -20(-2)^3 - 12(-2)^2 - 12(-2) = -160 - 48 + 24 = -184 ]

Since ( f''(-2) = -184 ) is negative, the function is concave at ( x = -2 ).

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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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