If y= (x^2+20)/(x-4) Find all critical Numbers.

After get f' i got the answer -5/4. What did i do wrong or is that the critical number?

To find the critical numbers of a function, you need to find the values of x where the derivative of the function is either zero or undefined.

To find the derivative of the given function y = (x^2 + 20)/(x - 4), you can use the quotient rule.

Start by finding the derivative of the numerator and denominator:

The derivative of the numerator (x^2 + 20) is 2x.
The derivative of the denominator (x - 4) is 1.

Using the quotient rule, you can find the derivative of y by following these steps:

Step 1: Multiply the derivative of the numerator by the denominator: (2x)(x - 4)
Step 2: Multiply the numerator by the derivative of the denominator: (x^2 + 20)(1)
Step 3: Subtract the result obtained in Step 2 from the result obtained in Step 1: (2x)(x - 4) - (x^2 + 20)(1)
Step 4: Simplify the above expression: 2x^2 - 8x - x^2 - 20
Step 5: Combine like terms: x^2 - 8x - 20

Now, to find the critical numbers, you need to set the derivative equal to zero and solve for x:

x^2 - 8x - 20 = 0

There are two ways you may have made a mistake:

1. Calculating the derivative incorrectly: Double-check your differentiation steps to ensure accuracy. Mistakes in derivative calculations can lead to incorrect critical numbers.

2. Misinterpreting the critical numbers: The critical numbers are the values of x where the derivative is either zero or undefined. In this case, -5/4 is not a critical number because it does not satisfy either condition.

To find the correct critical numbers, set the derivative equal to zero and solve the equation:

x^2 - 8x - 20 = 0

You can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Once you find the values of x that make the derivative zero, those will be the critical numbers.