# Thread: Maximum and minimum value of function?

1. ## Maximum and minimum value of function?

Can somebody please help me out with this question? I have done most of it and I just need to do the final part which is:

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Given the equation:

$\displaystyle y = \frac{a+bsinx}{b+asinx}$

where $\displaystyle 0<a<b$

Find the maximum and minimum values of y.

I can give you the following information I figured out and hopefully, that can help you help me:

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First, I discovered that the graph does NOT have any vertical asymptotes

Also, differentiating and simplifying it gave me $\displaystyle \frac{dy}{dx} = \frac{(b^2 - a^2)cosx}{(b+asinx)^2}$

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First, I believe that I may have to make $\displaystyle \frac{(b^2 - a^2)cosx}{(b+asinx)^2} = 0$

Now I'm stuck.

If you could help me out, it would be greatly appreciated. Thanks!

2. I'm going to assume your derivative calculation and simplification are correct. Then yes you are right the next step is to look where this expression equals 0 or is undefined. So,

The cosine function is periodic and will hit 0 an infinite amount of time. Generally $\displaystyle \cos(x)=0, x=\frac{k \pi}{2}$, where k is any odd integer.

The other non-trivial time f'=0 is when $\displaystyle b^2-a^2=0$ This means that either a=0 or one of the terms has to be negative. Since both cases are not possible, this solution doesn't hold with the constraints.

The derivative is undefined when $\displaystyle b+a\sin(x)=0$ (the squared doesn't matter here). So we're looking for when $\displaystyle \sin(x)= -\frac{b}{a}$ And since b>a always, that means b/a is greater than one and this has no solutions.

3. Thank you very much for your help