How to Find Critical Points of f(x) = xln(x)

In summary, the conversation is about finding the critical points of the equation f(x) = xln(x). The person has found the derivative to be ln(x)+x, but is unsure of how to solve for x when setting the derivative to 0. They mention being able to find the zeros graphically on a calculator, but need to know how to solve it algebraically. They then realize their mistake and thank the other person for their help.
  • #1
jtulloss
4
0
I need help finding the critical points of this equation:

f(x) = xln(x)

I found f'(x) to be ln(x)+x, but I don't know how to solve for x when setting f'(x) to 0 to find the critical points. I know I can find the zeros of f'(x) graphically on a calculator, but I need to know how to do it algebraically.

Thanks.
 
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  • #2
jtulloss said:
I need help finding the critical points of this equation:

f(x) = xln(x)

I found f'(x) to be ln(x)+x,
It's not lnx + x.
 
  • #3
Ah man, I can't believe I overlooked that. I've been doing that a lot lately with studying for exams and all. I got it now.

Thanks
 

What are critical points?

Critical points are points on a graph where the derivative is equal to zero or does not exist. They are important in finding maximum and minimum values of a function.

How do you find critical points?

To find critical points, you need to take the derivative of the function and set it equal to zero. Then, solve for the variable to find the x-values of the critical points. You can also use the second derivative test to determine whether the critical point is a maximum, minimum, or inflection point.

Why are critical points important?

Critical points are important because they help us find the maximum and minimum values of a function. They can also indicate where the function changes from increasing to decreasing or vice versa.

What is the significance of the second derivative test in finding critical points?

The second derivative test helps determine whether a critical point is a maximum, minimum, or inflection point. It involves taking the second derivative of the function and plugging in the x-value of the critical point. If the result is positive, the critical point is a minimum. If the result is negative, the critical point is a maximum. If the result is zero, the test is inconclusive.

Can there be more than one critical point for a function?

Yes, there can be more than one critical point for a function. This can occur when the function has multiple local maximum or minimum values. It can also happen when the function has an inflection point, where the slope changes from positive to negative or vice versa.

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