Certainly! One example of a function that is positive for the entire interval from 2 to 3 is the quadratic function:
f(x) = (x - 2.5)^2 + 0.1
This function is a parabola that opens upward, with its vertex located at (2.5, 0.1). Within the interval (2, 3), this function produces positive values because:
- At the left endpoint, when x = 2:
- f(2) = (2 – 2.5)^2 + 0.1 = (0.5)^2 + 0.1 = 0.25 + 0.1 = 0.35
- At the right endpoint, when x = 3:
- f(3) = (3 – 2.5)^2 + 0.1 = (0.5)^2 + 0.1 = 0.25 + 0.1 = 0.35
As you can see, both endpoints yield a positive value, and since the function is continuous and has no roots within the interval, it remains positive throughout. Additionally, any other function that maintains a positive value can be used, such as:
f(x) = e^{(x - 2)}
This exponential function will also yield positive results for any real number input and is consistently above zero in the entire interval from 2 to 3.