A) Finding Local Maximum and Local Minimum of a Function
To find the local maximum and local minimum of a function, we need to take the derivative of the function and set it equal to zero. We can then solve for x to find the critical points of the function.

1. Take the derivative of the function:
f(x) = x^4 - 62x^2 + 120x + 9
f'(x) = 4x^3 - 124x + 120

2. Set f'(x) equal to zero and solve for x:
4x^3 - 124x + 120 = 0
x = -1, 2, 15/2

3. Plug these x-values back into the original function to find the corresponding y-values:
f(-1) = 186
f(2) = -55
f(15/2) = 769/16

Therefore, the local maximum is at x = -1 with a y-value of 186, and the local minimum is at x = 15/2 with a y-value of 769/16.

B) Finding the Area of the Largest Rectangle with a Given Perimeter
To find the area of the largest rectangle with a given perimeter, we need to use the formula for the perimeter of a rectangle: P = 2(l + w), where l is the length and w is the width.

1. Rewrite the formula for the perimeter in terms of one variable:
P = 2(l + w)
P/2 = l + w
l = P/2 - w

2. Substitute this expression for l into the formula for the area of a rectangle:
A = lw
A = w(P/2 - w)
A = (P/2)w - w^2

3. Take the derivative of this function and set it equal to zero to find the critical point:
dA/dw = P/2 - 2w = 0
w = P/4

4. Plug this value of w back into the equation for the area to find the maximum area:
A = (P/2)(P/4) - (P/4)^2
A = P^2/16

Therefore, the area of the largest rectangle with a perimeter of P is P^2/16.

C) Dividing 30 into parts to maximize their product
To divide 30 into parts to maximize their product, we need to use the AM-GM inequality, which states that the arithmetic mean of a set of non-negative numbers is greater than or equal to the geometric mean of the same set of numbers.

1. Write the numbers as a product:
30 = x1 * x2 * x3 * ... * xn

2. Take the logarithm of both sides:
ln(30) = ln(x1) + ln(x2) + ln(x3) + ... + ln(xn)

3. Apply the AM-GM inequality:
ln(30) >= n * (ln(x1) * ln(x2) * ln(x3) * ... * ln(xn))^(1/n)

4. Simplify:
ln(30) >= n * ln[(x1 * x2 * x3 * ... * xn)^(1/n)]