Calc Question Maximizing

Question

Consider the function f(x)=7sin(x)−9cos(x). Functions like this come up frequently whenever some periodic behavior is being studied: sound waves, light waves, and electrical circuits are just a few examples. Graph the function on your calculator; you will see that it is similar to a sine wave, but shifted and expanded.

The maximum value of sin(x) is 1, and the same for cos(x). Find the maximum value of f(x).

[Hint: This is an optimization problem, but it is designed to trip you up a little if you are just blindly following procedures. To get the correct answer it will be important that you be looking at the graph of your function.]

Answer 1

To find minimums and/or maximums, take the derivative to find critical points:

f '(x) = 7cos(x) + 9sin(x) then set it equal to zero

7cos(x) + 9sin(x) = 0

7cos(x) = -9sin(x)

-7/9 = sin(x)/cos(x) [applicable only if cos(x) = 0 is not a solution, which it is not.

-7/9 = tan(x)

x = tan^-1(7/9)

x = 2.481 + 2πk where k is any integer

and x = 5.622 + 2πk where k is any integer

To determine if these are minimums and/or maximums, take the 2nd derivative:

f ''(x) = -7sin(x) + 9cos(x)

f ''(2.481) is negative therefore the function is concave down and the critical point is a maximum and this will occur at all other multiples so f(x) has a maximum at x = 2.481 + 2πk where k is any integer.

f ''(5.622) is positive meaning the function is concave up and x = 5.622 is a minimum.(as well as all multiples of it.

Perhaps I fell into the trap because I didn't see the trip-up unless it was the need to use the multiple maximum generator of + 2πk.