Calc stuff

Hi,

Thank you for your question.

First, we should be finding the critical points of the function (where the derivative of the function equals zero):

f ' (x) = { (2x)(x+2) - (x^2+3) } / ( (x+2)^2 )

f ' (x) = ( x^2 + 4x - 3 ) / ( (x+2)^2 )

First critical point you may see is x = -2, but this value is not in the domain, so ignore this.

Other critical points are when x^2 + 4x - 3 = 0. Using the quadratic equation, x = -2 + sqrt (7), -2 - sqrt (7).

Similarly, as -2 + sqrt(7) is not in the domain, the only relevant critical point is when x = -2 - sqrt (7).

Afterwards, we can check some points.

a) f ( -2 - sqrt (7) ) = -2sqrt(7) - 4

b) As x gets really close to -7 (right side limit), f(x) approaches negative infinity.

c) As x gets really close to -2 (left side limit), f(x) approaches negative infinity.

Therefore, there is no absolute minimum on the given interval. On the other hand, there is an absolute maximum of -2sqrt(7) - 4 when x = -2 - sqrt(7).

Does this answer your question?