SAT Mathematics Subject test problem

 Find the difference between the maximum and minimum values of  Cos 2x + Cos x.

Answer:

Let y = Cos 2x + Cos x

First derivative of y, when it equates to zero, gives maximum and/or minimum value of y.

dy/dx = d( Cos 2x + Cos x)/dx

         = -2 Sin 2x - Sin x

         = -4 Sin x . Cos x - Sin x

         = -Sin x (4 Cos x + 1)

Now, we equate dy/dx to zero and solve for x.

dy/dx = 0

=> -Sin x (4 Cos x +1) =0

Case 1:

Sin x =0

=> Cos x =1

=> x = 0

Therefore, Cos 2x + Cos x = Cos 0 + cos 0

                                       = 1 + 1

                                       = 2

This is maximum value of y.

Case 2:

4 Cos x + 1 = 0

 =>    Cos x = -1 / 4

=> x = 104.4775 degrees

Therefore, Cos 2x + Cos x = Cos (208.955) + Cos (104.4775)

                                       = (- 0. 875) + (-0.25)

                                       = - 1.125

Now, Maximum value of y is 2

        Minimum value of y is -1.125

Difference = 2 - (-1.125)

                = 2 + 1.125

                = 3.125