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Problem 1: Local Maximum of the Function $y = x^3 - 3x^2 - 4x + 12$

To find the local maximum, we need to:

1. Take the derivative of the function, $y'$, to find the critical points.
2. Set $y' = 0$ and solve for $x$.
3. Determine if these points are maxima by testing the second derivative or analyzing the sign of $y'$ around the points.

Step-by-Step Solution:

1. Differentiate $y$: $y = x^3 - 3x^2 - 4x + 12$ The derivative, $y'$, is: $y' = 3x^2 - 6x - 4$

2. Set the derivative equal to zero to find critical points: $3x^2 - 6x - 4 = 0$ Solve this quadratic equation using the quadratic formula: $x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot (-4)}}{2 \cdot 3}$ Simplifying this gives the values of $x$ at the critical points.

3. Evaluate $y$ at these points to find the function values and determine which is a maximum.

Problem 2: Determining if the Function is Even, Odd, or Neither

For $y = 20x^5 + 8x^4 - 2x^3$:

1. Check for an even function: A function is even if $f(-x) = f(x)$ for all $x$.
2. Check for an odd function: A function is odd if $f(-x) = -f(x)$ for all $x$.
3. Substitute $-x$ into the function to see if it matches either property.

Solution:

1. Substitute $-x$: $y(-x) = 20(-x)^5 + 8(-x)^4 - 2(-x)^3$ Simplify to check if it equals $y(x)$ or $-y(x)$.

If you would like further help with detailed calculations, please let me know!

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Here are five related questions:

1. How do you find inflection points for a cubic function?
2. What determines if a critical point is a maximum or minimum?
3. What is the general form for identifying even and odd functions?
4. How does the degree of a polynomial affect its symmetry?
5. Can a function be both even and odd?

Tip: For functions involving symmetry, always test $f(-x)$ for clear identification of even, odd, or neither.