Hi,
I am having a problem trying to find and understanding how to find the extreme values. I have a problem which I can't do. Will someone please explain for me how to do it. I would like to be able to do other similar problems. This is the problem I am stuck on: f(x)= x^4 – 4x^3 – x^2 + 12x – 2. Identify the extreme values.
Thank you in advance,
Cat
extreme values are where there is a maximum or minimum. At those points the curve changes direction, so it is neither increasing nor decreasing. Its slope is zero at those points.
So, how are you supposed to identify them? Graphically? Analytically? Since this is pre-cal, I assume you haven't yet learned about derivatives, so that makes things tough.
If you visit
http://rechneronline.de/function-graphs/
and enter your function, set the range (y-limits) to be -10 and 10, and you can see where the extrema are.
We are supposed to graph the equation then we are supposed to find the extreme values. I tried graphing it and I tried finding the extreme values. I came up with the max 5 and the min are -9 and -2. I am not sure if they are right because I watched a video online and didn't understand much of it. Thanks
I get:
max at f(1.14371) = 6.14328
min at f(-0.93854) = -10.0605
min at f(2.7948) = -2.58273
Hello Cat! I can definitely help you with finding the extreme values of a function. To find the extreme values, we need to find the critical points of the function, which are the points where the derivative is either zero or undefined. Once we have the critical points, we can check if they correspond to a local maximum or minimum by using the second derivative test or by analyzing the function's behavior on an interval.
Let's start with your problem: f(x) = x^4 – 4x^3 – x^2 + 12x – 2. To find the critical points, we need to find where the derivative is zero or undefined. So, let's start by finding the derivative f'(x). Differentiating each term of the function, we get:
f'(x) = 4x^3 - 12x^2 - 2x + 12.
Now, let's set f'(x) equal to zero and solve for x to find the critical points:
4x^3 - 12x^2 - 2x + 12 = 0.
Unfortunately, this equation is not easy to solve analytically, so we will need to use numerical methods or a graphing calculator to find the approximate solutions. If you have access to a graphing calculator or online graphing tool, you can graph the function f(x) and observe where the derivative is zero - those will be the critical points.
Since the derivative is a cubic equation, we expect to have at least one real solution. Once you have the approximate values for the critical points, you can substitute them back into the original function f(x) to find the corresponding extreme values.
In this case, finding the exact solutions for the critical points may not be practical, but you can use a graphing calculator or software to approximate the critical points. Once you have the x-values of the critical points, substitute them back into the original function f(x) to find the corresponding y-values. The highest y-value will be the maximum extreme value, and the lowest y-value will be the minimum extreme value.
I hope this explanation helps you understand how to find the extreme values of a function. Let me know if you have any further questions or need assistance with anything else!