Final answer:
The maximum value of the function -5.3sin(x) + 3.1cos(x) is 8.4, and the minimum value is -8.4.
Step-by-step explanation:
The function f(x) given by -5.3sin(x) + 3.1cos(x) is a linear combination of sine and cosine functions. To find the maximum and minimum values of this function, consider that sine and cosine functions oscillate between -1 and +1.
The maximum and minimum values of the function f(x) = -5.3sin(x) + 3.1cos(x) can be determined by analyzing the properties of the sine and cosine functions. The maximum value occurs when the sine term is at its maximum of 1 and the cosine term is at its minimum of -1.
So, the maximum value of f(x) is (-5.3)(1) + (3.1)(-1) = -5.3 - 3.1 = -8.4. Similarly, the minimum value occurs when the sine term is at its minimum of -1 and the cosine term is at its maximum of 1. So, the minimum value of f(x) is (-5.3)(-1) + (3.1)(1) = 5.3 + 3.1 = 8.4.
The maximum value of f(x) occurs when sin(x) equals -1 (because the sine term is multiplied by a negative coefficient) and cos(x) equals +1 since these will give the largest possible positive value for f(x). Conversely, the minimum value of f(x) occurs when sin(x) equals +1 and cos(x) equals -1. Thus:
- Maximum value: f(x) = -5.3(-1) + 3.1(1) = 5.3 + 3.1 = 8.4
- Minimum value: f(x) = -5.3(1) + 3.1(-1) = -5.3 - 3.1 = -8.4