HIGH SCHOOL STUDY HELP
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Please help me with this question :)
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?
Yes, thank you so much
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?