# What intervals is #f(x) = (5x^2)/(x^2 + 4)# concave up/down?

In order to investigate concavity, we shall look at the sign of the second derivative.

So:

Cut the number line into intervals and test each interval:

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To determine the intervals where the function ( f(x) = \frac{5x^2}{x^2 + 4} ) is concave up or concave down, we need to find the second derivative of the function and then analyze its sign.

Given ( f(x) = \frac{5x^2}{x^2 + 4} ), we first find the first derivative ( f'(x) ) using the quotient rule:

[ f'(x) = \frac{(10x(x^2 + 4) - 5x^2(2x))}{(x^2 + 4)^2} ]

Simplifying:

[ f'(x) = \frac{10x^3 + 40x - 10x^3}{(x^2 + 4)^2} ] [ f'(x) = \frac{40x}{(x^2 + 4)^2} ]

Now, to find the second derivative ( f''(x) ), we differentiate ( f'(x) ) with respect to ( x ):

[ f''(x) = \frac{40(x^2 + 4)^2 - 80x^2(x^2 + 4)(2x)}{(x^2 + 4)^4} ]

Simplifying:

[ f''(x) = \frac{40(x^4 + 8x^2 + 16) - 160x^3(x^2 + 4)}{(x^2 + 4)^3} ] [ f''(x) = \frac{40x^4 + 320x^2 + 640 - 160x^5 - 640x^3}{(x^2 + 4)^3} ] [ f''(x) = \frac{-160x^5 + 40x^4 - 640x^3 + 320x^2 + 640}{(x^2 + 4)^3} ]

Now, to determine the intervals where the function is concave up or concave down, we need to find the critical points of ( f''(x) ) and test the sign of ( f''(x) ) in those intervals.

However, since ( f''(x) ) is a polynomial, it has no critical points. Therefore, we'll need to analyze the sign of ( f''(x) ) by considering the behavior of ( x ) as it approaches positive and negative infinity.

As ( x ) approaches positive infinity, the dominant term in ( f''(x) ) is ( -160x^5 ), which is negative. Thus, ( f''(x) ) is negative for large positive ( x ) values, indicating that the function is concave down in that interval.

As ( x ) approaches negative infinity, the dominant term in ( f''(x) ) is also ( -160x^5 ), which is still negative. Thus, ( f''(x) ) is negative for large negative ( x ) values as well, indicating that the function is concave down in that interval.

Therefore, the function ( f(x) = \frac{5x^2}{x^2 + 4} ) is concave down for all real values of ( x ).

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