Consider the function f(x) = 3/4x^4 –x^3 -3x^2 + 6x , find the relative extrema for f(x), be sure to label each as a maximum or minimum.
How do I find the x-values of what it is asking?
better review your max/min topic. f has a max/min where f'=0 and f"≠0.
f' = x^3-3x^2-6x+6
This does not factor over the rationals, so a graphical solution is probably the best bet. The graph of f' is at
http://www.wolframalpha.com/input/?i=x^3-3x^2-6x%2B6
So, where f'=0, f will have a max or min. Just using what you know about the shape of quartics, you should be able to decide which are the maxes and which are the mins, without checking f".
To find the x-values of the relative extrema for the given function, you need to follow these steps:
Step 1: Find the derivative of the function f(x).
The derivative of f(x) will give you the slope of the function at any given point. To find the derivative, you need to use the power rule for differentiation. Take the derivative of each term in the function:
f'(x) = d/dx (3/4x^4) – d/dx (x^3) – d/dx (3x^2) + d/dx (6x)
Using the power rule, the derivative of each term can be calculated:
f'(x) = (4 * 3/4 * x^3) – (3 * x^2) – (2 * 3 * x) + 6
Simplifying further, we get:
f'(x) = 3x^3 - 3x^2 - 6x + 6
Step 2: Set the derivative f'(x) equal to zero and solve for x.
To find the x-values where the derivative is zero, we set f'(x) = 0 and solve for x:
3x^3 - 3x^2 - 6x + 6 = 0
This is a cubic equation, which can be solved using various methods such as factoring, synthetic division, or the cubic formula. Once you solve the equation, you will obtain the x-values of the critical points.
Step 3: Determine the type of each relative extremum.
To determine whether each critical point is a relative maximum or minimum, you need to examine the concavity of the function. One way to do this is by analyzing the second derivative of the function.
Find the second derivative f''(x) by taking the derivative of f'(x):
f''(x) = d/dx (3x^3 - 3x^2 - 6x + 6)
Differentiating each term, we get:
f''(x) = 9x^2 - 6x - 6
Now, substitute the x-values obtained from solving the equation in step 2 into the second derivative equation, and determine the sign of f''(x) at each point:
If f''(x) > 0, it indicates a relative minimum.
If f''(x) < 0, it indicates a relative maximum.
By determining the type of each point, you will be able to label them as either maximum or minimum relative extrema.