Let f(x)=4.1sin(x)+2.4cos(x). What is the maximum and minimum value of this function?

Question

Let f(x)=4.1sin(x)+2.4cos(x). What is the maximum and minimum value of this function?

 

 

 

 

note: I tried the last answer and it ended up being incorrect on the wamap system

 

 

 

 

Answer 1

Mark's answer is correct.  This is why.

If you rewrite the function, the answer becomes obvious.

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

Alternatively, to match what is below, sin(x + θ) = sin x cos θ + cos x sin θ

 

Thus, if we factor out sqrt(4.1^2 + 2.4^2), we may write 

4.1 sinx + 2.4 cosx as

 

sqrt(4.1^2+2.4^2) (sin x 4.1/sqrt(4.1^2+2.4^2)  + cos x 2.4/sqrt(4.1^2+2.4^2)  )

 

If cos θ = 4.1/sqrt(4.1^2+2.4^2) and sin = 2.4/sqrt(4.1^2+2.4^2), θ = tan^-1(2.4/4.1), and we have above

sqrt(4.1^2+2.4^2) (sin x cos (tan^-1(2.4/4.1)) + cos x sin(tan^-1(2.4/4.1))) = 

sqrt(4.1^2+2.4^2) sin (x + tan^-1(2.4/4.1))

 

As sin has a maximum of 1 and minimum of 1, the function has a maximum of sqrt(4.1^2+2.4^2) and a minimum of

-sqrt(4.1^2+2.4^2)

 

The maximum is sqrt(2257)/10, and the minimum is - sqrt(2257)/10

Note that 2257 = 61 * 37

Depending upon what answers are acceptable on your computer, you might rewrite this as 

2257^1/2/10 and - 2257^1/2/10

 

 

 

Answer 2

Let us solve this by setting dy/dx = 0, solving for x, and substituting x back into y = f(x):

 

f(x) = 4.1sin(x) + 2.4cos(x)

 

dy/dx = 4.1cos(x) - 2.4sin(x) = 0

 

4.1cos(x) = 2.4sin(x)

 

tan(x) = 4.1/2.4

 

x = tan^-1(4.1/2.4) = 59.6568º or (180 + 59.6568) = 239.6568º

 

At x = 59.6568º:

 

f(x) = 4.1sin(59.6568) + 2.4cos(59.6568) = 4.7508

 

At x = 239.6568º:

 

f(x) = 4.1sin(239.6568) + 2.4cos(239.6568) = -4.7508

 

f_max = 4.7508, f_min = -4.7508

 

I get same answer as Mark and Tom.

 

Answer 3

Using WolframAlpha.com

global max/min is ±√2257 / 10, or

±4.750

Answer 4

Since yours is a Trigonometry question, it would not seem appropriate to solve by Calculus methods.

Personally, I'd use a graphing calculator on the stated equation. Be sure that it's in RADIAN measure and set the window appropriately (say, x interval 0 to 2pi).

 

Now inspect the graph and ask the calculator to find the maximum and minimum in the locales that seem appropriate.

 

Answer 5

Take derivative, set equal to zero, and solve.

 

f'(x) = -4.3sin(x) - 6.7cos(x) = 0;

 

           -6.7/4.3 = sin(x)/cos(x) = tan (x) 

 

There are solutions of this equation in both the second and fourth quadrants.  Using a calculator reveals:

 

x = -1.0002135638   x = -1.0002135638 + π = 2.14137908979

 

Substitute these two values of x into the original equation to find the maximum and minimum for the function

 

f(-1.0002135638) = 7.23764264982  (max value)

f(2.14137908979) = -7.23764264982 (min value)