Find the Critical Numbers of the Function f(x) = x^8 – 8x + 37

To find the critical numbers of the function f(x) = x^8 – 8x + 37, we need to follow a systematic approach.

First, we start by calculating the first derivative of the function:

Step 1: Find the derivative

Using the power rule, we differentiate f(x):

f'(x) = 8x^7 - 8

Next, we set the derivative equal to zero to find critical points:

Step 2: Set the derivative to zero

8x^7 - 8 = 0

We can factor this equation:

8(x^7 - 1) = 0

Now, we solve for x:

x^7 - 1 = 0

This gives us:

x^7 = 1

Taking the seventh root of both sides, we find:

x = 1

So, we have one critical number at x = 1.

Step 3: Check for points where the derivative is undefined

The derivative, f'(x) = 8x^7 – 8, is a polynomial and thus is defined for all real x. Therefore, there are no additional critical numbers arising from where the derivative is undefined.

In summary, the only critical number for the function f(x) = x^8 – 8x + 37 is:

x = 1