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

You can use the *second derivative test*.

The inflexion points will determine on which intervals the sign of the second derivative should be examined.

So, start by calculating the function's first derivative - use the quotient rule

Next, calculate the second derivative - use the quotient and chain rules

This is equivalent to

Take the square root of both sides to get

graph{x^2/(x^2+1) [-4.932, 4.934, -2.465, 2.467]}

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To determine the concavity of the function ( f(x) = \frac{x^2}{x^2 + 1} ) and its intervals, we need to find the second derivative of the function and examine its sign.

The second derivative ( f''(x) ) will tell us whether the function is concave up or concave down. If ( f''(x) > 0 ), the function is concave up, and if ( f''(x) < 0 ), the function is concave down.

First, find the first derivative of ( f(x) ) with respect to ( x ): [ f'(x) = \frac{d}{dx} \left( \frac{x^2}{x^2 + 1} \right) ]

Then, find the second derivative ( f''(x) ): [ f''(x) = \frac{d^2}{dx^2} \left( \frac{x^2}{x^2 + 1} \right) ]

After finding ( f''(x) ), determine its sign to identify the intervals where the function is concave up or 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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