Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Find the maximum and minimum values of: $$ 4 \sin x + \frac{9}{1+\sin x}$$ For $ 0 \le x \le \pi $.

What I have done:

$$ \frac{dy}{dx} = 4\cos x-\frac{9\cos x}{(1+\sin x)^2}$$

After equating the above to $0$, I found that $ x=\pi/6 $.

I do not know what to do next to find the maximum and minimum values. Any help would be appreciated.

share|improve this question
    
The extreme values must occur at critical points or at endpoints of the interval. Calculate $f(\pi/6)$, $f(0)$, and $f(\pi)$, and compare the values. –  Brian M. Scott Jan 8 '13 at 18:19
    
Draw a picture first. The constraint $x \in [0,\pi]$ may be important, in which case you cannot just look for zeros of the derivative. –  copper.hat Jan 8 '13 at 18:19
    
If not sure on such questions it can be very useful to sketch a graph. –  Mark Bennet Jan 8 '13 at 18:19
    
@BrianM.Scott - you are assuming the function is continuous, which should probably be mentioned given the form it takes (ie note that discontinuities are outside the interval) –  Mark Bennet Jan 8 '13 at 18:22
1  
@Mark: Not to worry: I was actually more amused than annoyed. This might have been better conveyed by ‘No, I know that the function is continuous, and I’m assuming (possibly incorrectly) that the OP will check this’! –  Brian M. Scott Jan 8 '13 at 18:37

2 Answers 2

up vote 0 down vote accepted

First, notice that $\dfrac{\pi}{2}$ and $\dfrac{5\pi}{6}$ work as well. You can then use The Extreme Value Theorem. You merely plug in the critical values and the endpoints. The largest is the max, the smallest the min.

share|improve this answer

With apologies to Magritte, this is not an answer, but doesn't fit in the comments.

enter image description here

share|improve this answer
    
Thank you for the replies and yes I should have drawn a graph. But is there any method to work out the minimum and maximum values without the use of a graph? –  neverloggedin Jan 8 '13 at 18:24
    
See Brian's comment above. Either the constraints are relevant or not at a solution. If not, using the derivative can characterize a solution, but if a constraint is active, care must be taken. A trivial example is to maximize $x^3$ subject to $|x|=1$. Clearly the solution is $x=1$, but the derivative is of no help here. –  copper.hat Jan 8 '13 at 18:25
    
You remind me of my all-time favorite Usenet sig; see the bottom of this post. –  Brian M. Scott Jan 8 '13 at 18:28
    
@BrianM.Scott: Excellent! –  copper.hat Jan 8 '13 at 18:33

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.