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
Hi Cat!
Finding extreme values of a function involves finding the maximum and minimum points on the graph of the function. To do this, you need to find the critical points where the derivative of the function is equal to zero or undefined. Then, you evaluate the function at these critical points and compare the values to determine the extreme values.
Let's start by finding the critical points of the function f(x) = x^4 – 4x^3 – x^2 + 12x – 2.
1. Take the derivative of the function to find the critical points:
f'(x) = 4x^3 – 12x^2 – 2x + 12.
2. Set the derivative equal to zero and solve for x to find the critical points:
4x^3 – 12x^2 – 2x + 12 = 0.
Now, finding the roots of this polynomial equation can be challenging, but we can use numerical methods or technology to solve it. One possible approach is to use a graphing calculator or software to plot the graph of the function and see where it crosses the x-axis.
Alternatively, you can use factoring, the Rational Root Theorem, or synthetic division to find any possible rational roots of the function.
Once you find the critical points by solving the equation, you can evaluate the function at these points to determine if they correspond to a maximum or minimum. You can also check the behavior of the function at extreme values by analyzing the concavity of the graph using the second derivative test.
So, once you find the x-values of the critical points, substitute them back into the original function f(x) = x^4 – 4x^3 – x^2 + 12x – 2 to obtain the corresponding y-values.
By comparing the y-values, you can identify the maximum and minimum values of the function, which are the extreme values.
I hope this explanation helps you understand how to find the extreme values of a function. If you have any more questions or need further assistance, feel free to ask!