How Do You Find the Critical Numbers of a Function?

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In summary, critical numbers are points on a function where the derivative is equal to zero or undefined. They are found by taking the derivative of the function and setting it equal to zero. Critical numbers are important because they help us understand the behavior of a function, and a function can have more than one critical number. By using critical numbers, we can optimize a function for a specific goal, such as finding the maximum profit or minimum cost.
  • #1
helpm3pl3ase
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Find Critical Numbers of:

F(z) = (z+1)/(z^2+z+1)

Did I start off correctly??

D/dz = - (z + 1)(2z+1)/(z^2+z+1)

I think I derived it correctly? Where do I go from this step?? Thanks
 
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  • #2
D/dz = - (z + 1)(2z+1)/(z^2+z+1)
The derivative is wrong.

It should be

((z^2+z+1)-(z+1)(2z+1))/(z^2+z+1)^2
 
  • #3


Yes, you started off correctly by taking the derivative of the function F(z). To find the critical numbers, you need to set the derivative equal to zero and solve for z. So in this case, you would have:

-(z+1)(2z+1)/(z^2+z+1) = 0

Simplifying this, you get:

2z^2 + 3z + 1 = 0

Now you can use the quadratic formula to solve for z. The critical numbers will be the solutions to this equation.
 

1. What are critical numbers?

Critical numbers are points on a function where the derivative is equal to zero or undefined. They are also known as critical points or stationary points.

2. How do you find critical numbers?

To find critical numbers, you need to take the derivative of the function and set it equal to zero. Then, solve for the variable to determine the critical numbers.

3. Why are critical numbers important?

Critical numbers are important because they help us find important information about the behavior of a function, such as where it reaches a maximum or minimum value.

4. Can a function have more than one critical number?

Yes, a function can have more than one critical number. In fact, it is common for functions to have multiple critical numbers, especially when the function is complex.

5. How can critical numbers be used to optimize a function?

By finding the critical numbers of a function, we can determine where the function is increasing or decreasing, and where it reaches a maximum or minimum value. This information can be used to optimize the function for a specific goal, such as finding the maximum profit or minimum cost.

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