Question

Question is the attached file.

Answer 1

Answer:

The critical points are 12 and 0.

Step-by-step explanation:

We have that the critical numbers are those values that result from equating the derivative of a function to zero. Also called roots or zeros of the derived function.

IF f is defined in x, it will be said that a is a critical number of f if f '(x) = 0 or if f is not defined in x.

Now the function is:

f (x) = x ^ 2 / (x -6)

we have that the derivative of the quotient is:

(f / g) '= (f' * g - g '* f) / g ^ 2

we replace and we have:

f (x) = [2 * x * (x-6) - 1 * x ^ 2] / (x -6) ^ 2

simplifying we have:

f (x) = [x ^ 2 - 12 * x] / (x -6) ^ 2

this must be equal to 0, like this:

 [x ^ 2 - 12 * x] / (x -6) ^ 2 = 0

we solve:

x ^ 2 - 12 * x = 0

x * (x - 12) = 0

Thus:

x = 0

x - 12 = 0 => x = 12

The critical points are 12 and 0.