What are the critical values of f(x)=(x^(7))((x+7)^(2))

and
f(x)=7x+6x^(–1)?

To find the critical values of a function, we need to first find its derivative and then solve for the values of x where the derivative is equal to zero or undefined.

Let's start by finding the derivative of the first function, f(x) = (x^7)((x+7)^2).

1. Apply the product rule to differentiate the function:
f'(x) = (7x^6)((x+7)^2) + (x^7)(2(x+7)).

Simplifying further, we have:
f'(x) = 7x^6(x+7)^2 + 2x^7(x+7).

Next, we can find the derivative of the second function, f(x) = 7x + 6x^(-1).

2. Take the derivative of each term separately:
f'(x) = 7 + 6(-1)x^(-2).

Simplifying further, we have:
f'(x) = 7 - 6/x^2.

Now, let's solve for the critical values by setting the derivatives equal to zero and solving for x.

For the first function:
7x^6(x+7)^2 + 2x^7(x+7) = 0.

For the second function:
7 - 6/x^2 = 0.

Solving these equations may be challenging analytically, but we can use numerical methods or graphing calculators to find the critical values.

By finding the solutions to these equations, you will obtain the critical values for the given functions.