# How do you determine whether the function #f(x)= 4/(x^2+1)# is concave up or concave down and its intervals?

So let us calculate:

graph{4/(x^2+1) [-10, 10, -5, 5]}

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To determine the concavity of the function ( f(x) = \frac{4}{x^2 + 1} ) and its intervals, you need to find its second derivative. Then, if the second derivative is positive, the function is concave up, and if it's negative, the function is concave down. The concavity can change at points where the second derivative is zero or undefined. Therefore, to find the intervals of concavity, you can identify the intervals where the second derivative is positive (concave up) or negative (concave down).

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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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