use the derivative f'(x)= 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 critical points by solving the equation f'(x) = 0.
Step 2: Determine the nature of each critical point using the second derivative test.
Step 3: Find the inflection points by solving the equation f''(x) = 0 or by analyzing the behavior of f''(x) around the points where it changes sign.
Let's start with the given derivative:
f'(x) = x^4 - 1
Step 1: Find the critical points by solving f'(x) = 0.
To find where f'(x) = 0, set x^4 - 1 = 0 and solve for x:
x^4 - 1 = 0
(x^2 - 1)(x^2 + 1) = 0
x^2 - 1 = 0 or x^2 + 1 = 0
Solving each equation separately:
For x^2 - 1 = 0:
x^2 = 1
x = ±√1
x = ±1
For x^2 + 1 = 0:
This equation has no real solutions.
So, the critical points are x = -1 and x = 1.
Step 2: Determine the nature of each critical point using the second derivative test.
To do this, we need to find the second derivative of the original function.
First, let's find the first derivative of f'(x):
f''(x) = (x^4 - 1)' = 4x^3
Next, substitute the critical points into f''(x) to determine the nature of each critical point.
For x = -1:
f''(-1) = 4(-1)^3 = -4
Since f''(-1) < 0, the point x = -1 is a local maximum.
For x = 1:
f''(1) = 4(1)^3 = 4
Since f''(1) > 0, the point x = 1 is a local minimum.
Step 3: Find the inflection points by solving f''(x) = 0 or by analyzing the behavior of f''(x) around the points where it changes sign.
To find the inflection points, we solve the equation f''(x) = 0:
4x^3 = 0
x^3 = 0
x = 0
At x = 0, f''(x) = 0. So, x = 0 is a possible inflection point.
To analyze the behavior of f''(x) around the points where it changes sign, we can use test values. Since f''(x) = 4x^3, taking values less and greater than 0 can help us analyze the sign of f''(x).
For x < 0 (e.g., x = -2):
f''(-2) = 4(-2)^3 = -32 < 0
For x > 0 (e.g., x = 2):
f''(2) = 4(2)^3 = 32 > 0
Since f''(x) changes sign from negative to positive at x = 0, we can conclude that x = 0 is an inflection point.
In summary, the local extrema are:
- Local maximum at x = -1
- Local minimum at x = 1
The inflection point is at x = 0.