The equation $f(x) = ax^3 + bx^2 + cx + d$, where $a \neq 0$ represents a cubic function (Polynomial of degree 3).

a.) Prove that a cubic function can have one, two or more critical number(s). Give examples and sketches to illustrate the three possibilities.

$f(x) = ax^3 + bx^2 + cx + d$

Taking the derivative,

$f'(x) = 3ax^2 + 2bx + c$

We know that $f'(x)$ is a quadratic function that might have either $0, 1$ or $2$ real roots. This means that $f(x)$ has either $0, 1$ or $2$ critical numbers.

For 1 critical number:

Let $f(x) = x^3$, then $f'(x) = 3x^2$, thus $x = 0$ is the only critical number.

For 2 critical numbers:

Let $f(x) = 4x^3 - 12x$, then $f'(x) = 12x^2 - 12$, thus $x = \pm 1$ are the two critical numbers.

For 0 critical number:

Let $f(x) = 4x^3 + 12x$, then $f'(x) = 12x^2 + 12$, there are no real roots, thus, no critical numbers.

b.) How many local extreme values can a cubic function have?

If the function has one or no critical values, it has no extremum.

But if the function has 2 critical values, it can have at most 2 extrema.

Therefore, a cubic function can have either 2 extrema or none.

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