Find the critical number(s) of f(x)= abs(2x-4)
What is the derivative of abs(2x-4)?
Is it abs(2)?
No. You have to specify the domain of x. At x=2, this function has no derivative.
At x>2, f' = d (2x -4)/dx= 2
at x <2, f'= d(4-2x)/dx= -2
You have to watch for discontinuities on abs functions.
To find the critical numbers of the function f(x)= abs(2x-4), we need to find the values of x where the derivative of the function does not exist or is equal to zero.
To find the derivative of abs(2x-4), we need to consider the different cases based on the domain of x:
Case 1: x > 2
In this case, the absolute value function can be simplified to 2x-4, as the expression (2x-4) is positive. Taking the derivative of 2x-4 with respect to x gives us 2.
Case 2: x < 2
In this case, the absolute value function can be simplified to -(2x-4), as the expression (2x-4) is negative. Taking the derivative of -(2x-4) with respect to x gives us -2.
However, there is a discontinuity at x = 2 for the function abs(2x-4). This means that the derivative does not exist at x = 2.
Therefore, the critical numbers of f(x)= abs(2x-4) are x = 2 and x > 2 (where the derivative is equal to zero), and x < 2 (where the derivative is equal to zero).