I know how to calculate critical points using the derivative, but on this question I don't even know where to start. HELP!
It is the final exam period and you have only math and chemistry exams left. They are on the same day and you have 20 hours of study time remaining. You estimate that your mark on the chemistry exam as a function of hours spent studying is
$$C(x)=100\dfrac{x}{4+x}$$
You can't be quite as certain about your mark in your math course but you know it has a similar form:
$$M(x)=100\dfrac{x}{k+x}$$
where: $$k>0$$"
Write down a function for your average mark A(x) on the two exams where x is the number of hours spent studying math. Assume the remaining time is spent entirely on chemistry. Write your answer in terms of the functions C and M.
I think this is asking for A(x) where x is hours on domain 0< x <20. So time spent on math is x and time spent on chem is 20-x. So to average the functions I wrote $$A(x)=\dfrac{{C(20-x)}+{M(x)}}{2}$$.
Verify by calculating them from scratch that there are two critical points of A(x) given by $$x_1=\dfrac{2\sqrt{k}(12-\sqrt{k})}{\sqrt{k}+2}$$ and $$x_2=\dfrac{2\sqrt{k}(12+\sqrt{k})}{\sqrt{k}-2}$$. Hint: it is easiest to proceed as far as possible with $$A(x) written in terms of C and M before substituting in the algebraic expressions for these functions (or more accurately, the algebraic expressions for their derivatives).
I think this is asking for the first and second derivative of the function A(x) found above, but I have no idea how to take the derivative of the function I made, so I'm suspecting I made the wrong function.
- For what values of k is x1 in the domain of the model? For what values of k is x2 in the domain of the model?
Totally stuck here