Question

Let f be the function given by f(x) = sin(x)e⁽⁻ˣ⁾3x. Which of the following statements is true for y = f(x)?

1) y is a constant function
2) y is an increasing function
3) y is a decreasing function
4) y is a periodic function

Answer 1

The function f(x) = sin(x)e⁽⁻ˣ⁾3x is a decreasing function.

Step-by-step explanation:

The function f(x) = sin(x)e⁽⁻ˣ⁾3x can be analyzed to determine its properties. The function involves the product of two functions: sin(x) and e⁽⁻ˣ⁾3x. The sine function, sin(x), is periodic with a period of 2π and oscillates between -1 and 1. The exponential function, e⁽⁻ˣ⁾3x, decreases exponentially as x increases. Therefore, the function f(x) will exhibit both periodic behavior and a decreasing trend with increasing x. Thus, the correct statement for the function y = f(x) is 3) y is a decreasing function.