f ( x ) = − 4.5 sin ( x ) − 4.6 cos ( x ) . What is the maximum and minimum value of this function?

Answer 1

The maximum occurs when sin x and cos x are both < 0

The maximum occrus when sin x = cos x

The maximum occurs when tan x = 1

Interestingly the minimum also occrus when tan x = 1, i.e., sin x and cos x > 0

Answer 2

The minimum/maximum values will occur where the derivative of the function is zer0

so start by taking the derivative of the function

f '(x) = -4.5cosx -4.6(-sinx) = 4.6sinx -4.5cosx which I will set to equal zero

4.6 sinx -4.5 cosx=o

4.6sinx = 4.5cosx

sinx/cosx = 4.5/4.6 sin/cos is the same as tangent

tanx = 4.5/4.6 to solve for x take arc tan of both sides

x= .774radians +( n)pi since the period of tangent is pi These will alternate between minima and maxima every pi radians So Minimum is at .7744 rad and maximum is at .7744 +pi= 3.9159 rad

These will repeat indefinitely so answer is .7744+ n(2pi) and 3.9159+n(2pi)