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Determine the intervals where the function g(x)= (x)/(x+6) is concave upward.

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Final answer:

To determine the intervals where the function g(x) = (x)/(x+6) is concave upward, find the second derivative of the function and set it greater than zero.

Step-by-step explanation:

To determine the intervals where the function g(x) = (x)/(x+6) is concave upward, we need to find the second derivative of the function and set it greater than zero.

First, let's find the first derivative of g(x) using the quotient rule: g'(x) = [(x+6)(1) - (x)(1)] / (x+6)^2 = 6 / (x+6)^2.

Next, let's find the second derivative of g(x) using the quotient rule again: g''(x) = [(x+6)^2(0) - (6)(2)(x+6)] / (x+6)^4 = -12 / (x+6)^3.

Now, set g''(x) > 0 to find the intervals where g(x) is concave upward: -12 / (x+6)^3 > 0. Since the numerator is negative, the inequality is true when the denominator is positive, which is X < -6.

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