Question

For each interval, indicate whether the function is increasing, decreasing, or neither increasing nor decreasing. x is between 0 and 2. x is between 2 and 4 x is between 4 and 6 x is between 6 and 8 x is between 8 and 11 x is between 11 and 12 Increasing Decreasing Neither increasing nor Decreasing​

Answer 1

The function f(x)=x^2 −x−1 is neither strictly increasing nor strictly decreasing on the interval (0,1).

To determine if a function f(x) is increasing or decreasing on an interval, we typically look at the derivative of the function.

The function f(x)=x^2 −x−1 can be differentiated with respect to x to find its derivative:

f′ (x)= d/dx (x^2 −x−1)

f′ (x)=2x−1

Now, to determine whether the function is increasing or decreasing on the interval (0,1), we can analyze the sign of its derivative within that interval.

For x∈(0,1):

Choose a value of x in the interval, say x= 1/2

Plug this value into the derivative:

f′ ( 1/2 )=2× 1/2 −1=0.

The derivative is zero at x= 1/2 .

This means the function f(x)=x^2 −x−1 has a critical point at x= 1/2

To determine whether it's increasing or decreasing, we can examine the sign of the derivative around this critical point by picking values on either side of it.

For x=0.3:

f′ (0.3)=2×0.3−1=−0.4

For x=0.7:

f′ (0.7)=2×0.7−1=0.4

The derivative is negative when x=0.3 and positive when x=0.7, indicating that the function changes from decreasing to increasing at x= 1/ 2

​within the interval (0,1). Therefore, the function f(x)=x^2 −x−1 is neither strictly increasing nor strictly decreasing on the interval (0,1).

Question

Show that the function x^2-x-1 is neither increasing nor decreasing on (0, 1).