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).