Tips for Max/Min Problems

Your assignment is p.229/43, 45, 46. 
43.)  Already discussed.  If the minimum of a function is positive, can the function ever be negative?  Find the exact minimum.

45.)  a.)  At the moment that the springs have the same position, they are passing each other.  No calculus required - just solve a trig equation.  You will have to use the identity in the hint.  Remember that, to solve trig equations, a common strategy is to get all terms to one side (set = 0) and factor out the GCF. 

b.)  Calculus required.  You have to maximize the distance between them.  SO, you must find an equation for that distance.  You should be tempted to use absolute values here but you don't have to.  Consider this "simple sample":

The points -3, -2, 0, 1 and 5 are on a number line and you want to know the greatest distance between any two numbers.  If you subtract in one direction, you get -3 - 5 = -8.  If you subtract 5 - (-3), you get 8.  The maximum distance is 8 and the minimum distance is its opposite.  Maximizing (or minimizing) will still get you that -3 and 5 are the farthest apart.

Same with this spring problem.  Subtract the positions to find distance, m1 - m2.  Differentiating and finding the 0's will give you a max, a min, or both.  Use the First Derivative Test.  When you find the actual distances by plugging the critical values back in, ignore any negative signs in the distance values.

If you have trig questions, post them and I'll check back later.

46.)  a.)  Same as the last one.  Set up and solve an equation.  Remember the Sum Identity:

sin (a + b) = sin a cos b + cos a sin b

You should get an equation that looks something like.....  b sin t = d cos t  (b, d are numbers).  Divide both sides by (b cos t) so that you can solve a tangent equation. 

b.)  Also, same as above.  Subtract positions and call that D (distance between them).  Differentiate D.  Find 0's.  First Derivative Test.  Plug values of t back in to find greatest distance.

c.)  When is the distance between the particles changing the fastest??  I don't want to spoil the satisfaction of figuring this out.  What are you trying to maximize?  What equation must be differentiated?