Hi, I needed some help with this question.
The model
. . . . .\(\displaystyle g(x)\, =\, \dfrac{bc\, +\, ax^d}{c\, +\, x^d}\)
is a saturation kinetics model used to describe the weight gain of chickens, \(\displaystyle g(x),\) with respect to the concentration \(\displaystyle x\) of lysine in their diet. Here \(\displaystyle d\, <\, b\, <\, a\, <\, c\) are positive constants, and the domain is restricted to \(\displaystyle x\, >\, 0.\)
(a) Determine \(\displaystyle \displaystyle \lim_{x \rightarrow 0^+}\, g(x)\) and \(\displaystyle \displaystyle \lim_{x \rightarrow \infty}\, g(x)\)
(b) Determine where \(\displaystyle g(x)\) is increasing and where it is decreasing.
(c) Determine where \(\displaystyle g(x)\) is concave up and where it is concave down.
(d) Draw a large sketch of the graph of \(\displaystyle g(x),\) indicating all the information determined in the previous parts of this question.
(e) Explain in one or two sentences a limitation of this model. Your answer should refer both to the mathematical features of the model and to the physical phenomenon the model is describing.
I have figured out the solution for part a, but I am having a great deal of trouble with part b and c. I have tried applying the quotient rule to the function that is given, but I am unsure if what I am doing is even correct. Any help or examples are greatly appreciated.
The model
. . . . .\(\displaystyle g(x)\, =\, \dfrac{bc\, +\, ax^d}{c\, +\, x^d}\)
is a saturation kinetics model used to describe the weight gain of chickens, \(\displaystyle g(x),\) with respect to the concentration \(\displaystyle x\) of lysine in their diet. Here \(\displaystyle d\, <\, b\, <\, a\, <\, c\) are positive constants, and the domain is restricted to \(\displaystyle x\, >\, 0.\)
(a) Determine \(\displaystyle \displaystyle \lim_{x \rightarrow 0^+}\, g(x)\) and \(\displaystyle \displaystyle \lim_{x \rightarrow \infty}\, g(x)\)
(b) Determine where \(\displaystyle g(x)\) is increasing and where it is decreasing.
(c) Determine where \(\displaystyle g(x)\) is concave up and where it is concave down.
(d) Draw a large sketch of the graph of \(\displaystyle g(x),\) indicating all the information determined in the previous parts of this question.
(e) Explain in one or two sentences a limitation of this model. Your answer should refer both to the mathematical features of the model and to the physical phenomenon the model is describing.
I have figured out the solution for part a, but I am having a great deal of trouble with part b and c. I have tried applying the quotient rule to the function that is given, but I am unsure if what I am doing is even correct. Any help or examples are greatly appreciated.
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