Make a singn diagram for the derivative of the function f(x)= (x-1)^4
To create a sign diagram for the derivative of the function f(x) = (x-1)^4, we need to find the critical points and the intervals where the derivative is positive or negative.
Step 1: Find the derivative.
To find the derivative of f(x), we can apply the power rule for differentiation. The power rule states that for a function g(x) = x^n, the derivative of g(x) with respect to x is given by g'(x) = nx^(n-1).
Using this rule, we can find the derivative of f(x) = (x-1)^4 as follows:
f'(x) = 4(x-1)^(4-1) = 4(x-1)^3
Step 2: Find the critical points.
Critical points occur when the derivative is equal to zero or undefined. To find the critical points, set the derivative equal to zero and solve for x:
4(x-1)^3 = 0
Since the derivative is in the form of a cube, it can only equal zero when the expression inside the parentheses equals zero. So we solve for x-1:
(x-1)^3 = 0
By taking the cubed root of both sides, we get:
x-1 = 0
Solving for x, we find the critical point:
x = 1
So the critical point is x = 1.
Step 3: Determine the intervals of positivity and negativity.
To determine the intervals where the derivative is positive or negative, we need to choose test points within each interval and evaluate the sign of the derivative at those points.
Choose a test point in the interval (negative infinity, 1) and plug it into the derivative:
For x = 0, f'(0) = 4(0-1)^3 = -4
So the derivative is negative in the interval (negative infinity, 1).
Choose a test point in the interval (1, positive infinity) and plug it into the derivative:
For x = 2, f'(2) = 4(2-1)^3 = 4
So the derivative is positive in the interval (1, positive infinity).
Finally, we can create the sign diagram:
```
-∞ 1 +∞
----------------|------------------
(-) ( + )
```
The interval (negative infinity, 1) represents where the derivative is negative, and the interval (1, positive infinity) represents where the derivative is positive.
Therefore, the sign diagram for the derivative of f(x) = (x-1)^4 is:
```
-∞ 1 +∞
----------------|------------------
(-) ( + )
```
Note: The critical point x = 1 is not included in the sign diagram because it does not affect the sign of the derivative.