How to find if a function #f(x)= 4/(x^2+1)# is concave up or concave down?

Question

Aurora Alvarado · Accepted Answer

You must study the sign of the second derivative as:

Luke Alvarez · Answer

undefined To determine if the function $ f(x) = \frac{4}{x^2 + 1} $ is concave up or concave down, you need to find the second derivative of the function and then examine its sign.

First, find the first derivative of $ f(x) $:
$$ f'(x) = \frac{d}{dx} \left( \frac{4}{x^2 + 1} \right) $$

Then, find the second derivative:
$$ f''(x) = \frac{d^2}{dx^2} \left( \frac{4}{x^2 + 1} \right) $$

After finding $ f''(x) $, determine its sign. If $ f''(x) > 0 $ for all $ x $ in the domain, the function is concave up. If $ f''(x) < 0 $ for all $ x $ in the domain, the function is concave down.

Axel Alvarez · Answer

undefined To determine if the function $ f(x) = \frac{4}{x^2 + 1} $ is concave up or concave down, we need to find the second derivative of the function and then analyze its sign.

1. Find the first derivative of the function $ f(x) $ with respect to $ x $.
   
2. Once you have the first derivative, find the second derivative by differentiating the first derivative with respect to $ x $.

3. After obtaining the second derivative, determine the intervals where the second derivative is positive and where it is negative.

- If the second derivative is positive on an interval, the function is concave up on that interval.
   - If the second derivative is negative on an interval, the function is concave down on that interval.

Analyzing the sign of the second derivative will give us information about the concavity of the function across its domain.

It's worth noting that the function $ f(x) = \frac{4}{x^2 + 1} $ is continuous and smooth for all real values of $ x $, so we can apply the second derivative test without any restrictions.

Once you have determined the concavity of the function, you can interpret the results in the context of the graph of the function. Concave up sections of the graph will have a "U" shape, while concave down sections will have an inverted "U" shape.