Use the derivative f'(x0= x^4-1 to find the local extrema and inflection points of f.
To find the local extrema and inflection points of a function, we need to follow a few steps:
Step 1: Find the derivative of the function.
Step 2: Set the derivative equal to zero and solve for x. These values of x will give us the critical points.
Step 3: Use the Second Derivative Test to determine the nature of each critical point.
Step 4: Find the points of inflection by checking the concavity of the function.
Let's go through these steps one by one.
Step 1: Find the derivative of the function.
The given function is f(x) = x^4 - 1. To find the derivative, we differentiate the function with respect to x.
f'(x) = 4x^3
Step 2: Set the derivative equal to zero and solve for x.
To find the critical points, we set the derivative equal to zero and solve for x.
4x^3 = 0
By factoring, we get:
4x^3 = 0
x^3 = 0
The only solution to this equation is x = 0.
Therefore, x = 0 is the only critical point of the function.
Step 3: Use the Second Derivative Test to determine the nature of the critical point.
To classify the critical point at x = 0, we need to find the second derivative.
f''(x) = 12x^2
Substituting x = 0 into the second derivative, we get:
f''(0) = 12(0)^2
f''(0) = 0
Since the second derivative is zero at x = 0, the Second Derivative Test is inconclusive. We need to use another method to determine the nature of the point.
Step 4: Find the points of inflection by checking the concavity of the function.
To find the inflection points, we need to check the concavity of the function. This can be done by analyzing the sign of the second derivative.
For a point to be an inflection point, the sign of the second derivative should change on either side of the point.
Since the second derivative is f''(x) = 12x^2, we can see that for x < 0, the second derivative is positive, and for x > 0, the second derivative is also positive.
Since the sign of the second derivative does not change, there are no inflection points in this case.
To summarize:
- The critical point is x = 0.
- The nature of the critical point cannot be determined by the Second Derivative Test.
- There are no inflection points in this function.
Please note that this is the analytical approach to finding local extrema and inflection points. You can also visualize the function or use technology to better understand the graph and locate the extrema and inflection points.