Which points are the best approximations of the relative maximum and minimum of the function?
f(x) = x3 + 3x2 – 9x –8
(1 point)
Responses
The relative maximum is at (3, –13), and the relative minimum is at (–3, –19).
The relative maximum is at (3, –13), and the relative minimum is at (–3, –19).
The relative maximum is at (–3, 19), and the relative minimum is at (1, –13).
The relative maximum is at (–3, 19), and the relative minimum is at (1, –13).
The relative maximum is at (3, –13), and the relative minimum is at (3, –19).
The relative maximum is at (3, –13), and the relative minimum is at (3, –19).
The relative maximum is at (–3, –19), and the relative minimum is at (–1, –13).
The relative maximum is at (–3, 19), and the relative minimum is at (1, –13).
The best approximations of the relative maximum and minimum of the function f(x) = x^3 + 3x^2 - 9x - 8 are:
- The relative maximum is at (3, -13)
- The relative minimum is at (-3, -19)
To find the relative maximum and minimum of a function, you need to analyze its critical points and determine the sign changes in the derivative.
First, let's find the derivative of the function f(x):
f'(x) = 3x^2 + 6x - 9
Next, set the derivative equal to zero and solve for x to find the critical points:
3x^2 + 6x - 9 = 0
By factoring or using the quadratic formula, you can find that the solutions are x = -3 and x = 1.
Now, we need to determine the sign changes in the derivative around these critical points in order to identify the relative maximum and minimum.
For x < -3, pick a test point, such as x = -4, and substitute it into the derivative:
f'(-4) = 3(-4)^2 + 6(-4) - 9 = -15
Since the derivative is negative in this interval, the function is decreasing.
For x between -3 and 1, take another test point, such as x = 0, and substitute it into the derivative:
f'(0) = 3(0)^2 + 6(0) - 9 = -9
Again, the derivative is negative, so the function is still decreasing.
For x > 1, take a final test point, such as x = 2:
f'(2) = 3(2)^2 + 6(2) - 9 = 15
This time, the derivative is positive, indicating that the function is increasing.
Based on this information, we can conclude that there is a relative maximum at x = -3 and a relative minimum at x = 1.
Substituting these x-values back into the original function, we find the corresponding y-values:
f(-3) = (-3)^3 + 3(-3)^2 - 9(-3) - 8 = -19
f(1) = (1)^3 + 3(1)^2 - 9(1) - 8 = -13
Therefore, the correct answer is: The relative maximum is at (-3, -19), and the relative minimum is at (1, -13).