# Is #f(x)=5x^3-2x^2+5x+12# concave or convex at #x=-1#?

The function is concave at

A concave function is a function in which no line segment joining two points on its graph lies above the graph at any point.

A convex function, on the other hand, is a function in which no line segment joining two points on the graph lies below the graph at any point.

graph{5x^3-2x^2+5x+12 [-2, 2, -20, 20]}

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To determine if a function is concave or convex at a given point, you need to look at the second derivative of the function at that point. If the second derivative is positive, the function is concave up (convex) at that point. If the second derivative is negative, the function is concave down (concave) at that point.

In this case, the function is f(x) = 5x^3 - 2x^2 + 5x + 12. The first derivative is f'(x) = 15x^2 - 4x + 5. The second derivative is f''(x) = 30x - 4.

At x = -1, the second derivative is f''(-1) = 30(-1) - 4 = -30 - 4 = -34, which is negative. Therefore, the function is concave down (concave) at x = -1.

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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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- What are the points of inflection, if any, of #f(x)=4x^3 + 21x^2 - 294x +7 #?

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