1. State whether f (x) = x^-1 has a minimum value, maximum value, or neither on the following intervals. Select all that apply.
(a) (0, 3)
(b) (1, 3)
(c) [1, 3)
Need Help.
f(x)=1/x
f'(x)=-1/x²
Since f'(x) does not have a real root, it does not have a maximum, nor a minimum.
Can you take it from here?
To determine whether the function f(x) = x^(-1) has a minimum value, maximum value, or neither on the given intervals, we can analyze the behavior of the function on each interval.
(a) (0, 3):
In this interval, the function includes all values between 0 and 3, excluding the endpoints. To identify if the function has a minimum or maximum value, we need to check if the derivative of the function is equal to zero or undefined.
First, let's find the derivative of f(x) = x^(-1). Using the power rule, we have:
f'(x) = -1 * x^(-1-1) = -1 * x^(-2) = -1/x^2
The derivative is defined for all x ≠ 0. So, there are no potential minimum or maximum values on the interval (0, 3).
(b) (1, 3):
Similar to the previous interval, we need to check if the derivative is equal to zero or undefined. Let's find the derivative:
f'(x) = -1/x^2
Again, the derivative is defined for all x ≠ 0. Thus, there are no potential minimum or maximum values on the interval (1, 3).
(c) [1, 3):
This interval includes the endpoints 1 and 3. To determine if the function has a minimum or maximum value, we will compare the function values at the endpoints and the critical points (where the derivative is equal to zero or undefined).
Let's find the derivative:
f'(x) = -1/x^2
The derivative is defined for all x ≠ 0. Therefore, the function f(x) = x^(-1) does not have any critical points within the interval [1, 3].
Now, let's evaluate the function at the endpoints:
f(1) = 1^(-1) = 1
f(3) = 3^(-1) = 1/3
The function values at the endpoints are different, indicating that there are no minimum or maximum values within the interval [1, 3].
In conclusion, for the given intervals:
(a) (0, 3): Neither minimum nor maximum value.
(b) (1, 3): Neither minimum nor maximum value.
(c) [1, 3): Neither minimum nor maximum value.