nesan said:Homework Statement
http://www.wolframalpha.com/input/?i=((1+/+x)+++2)+/+(3(2x+++lnx))^(2/3)+=+0
Solve the equation f'(x) = 0
f(x) = (ln x + 2x)^(1/3)
The Attempt at a Solution
As you can see, for the function
((1 / x) + 2) / (3(2x + lnx))^(2/3) = 0
I can get a value for x which is - 1 / 2
But my textbook says "no solution"
Why is that?
Mark44 said:x must be positive in order for ln(x) to be defined.
Solving f'(x) = 0 allows us to find the critical points of a function, which are points where the slope of the function is either zero or undefined. These points are important in determining the behavior of the function and can help us find the maximum and minimum values of the function.
Yes, there can be more than one solution to f'(x) = 0. This means that there can be multiple critical points on the function where the slope is either zero or undefined.
If there is no solution to f'(x) = 0, it means that there are no critical points on the function. This could indicate that the function is either constantly increasing or decreasing, or that it has no maximum or minimum values.
The solutions to f'(x) = 0 can be used to find the x-coordinates of the critical points on the function. These points can then be plotted on a graph to help visualize the behavior of the function and determine its maximum and minimum values.
Yes, there are other methods such as using the first and second derivative tests or using technology such as graphing calculators. These methods can also help us find the critical points of a function and determine its behavior.