Consider the cubic graph y = 3x^2 − x^3.
(a) Write 3x^2 − x^3 in factored form.
(b) Use this form to explain why the graph lies below the x-axis only when x > 3, and why the origin is therefore an extreme point on the graph.
I mostly need help with what an extreme point is,
Thanks!
(a) Sure, let's factorize the cubic equation y = 3x^2 − x^3.
We can start by factoring out an x^2 term:
y = x^2(3 - x)
Next, we notice that (3 - x) is the difference of squares, so we can factorize it further:
y = x^2(√(3 - x) * √(3 - x))
Therefore, in factored form, y = x^2(√(3 - x))^2.
(b) An extreme point is a point on a graph where the derivative is zero, indicating a potential maximum or minimum. In this case, the origin (0, 0) is an extreme point because the graph lies below the x-axis only when x > 3.
To understand why, let's analyze the factors of the factored form. We have x^2, which is always positive or zero. Then we have (√(3 - x))^2, which is also always positive or zero since we are squaring it.
If x > 3, then 3 - x will be negative, making (√(3 - x))^2 positive and the overall expression y = x^2 * (√(3 - x))^2 positive. Therefore, the graph will be above the x-axis for x > 3.
On the other hand, if x < 3, then 3 - x will be positive, resulting in (√(3 - x))^2 being positive, but since x^2 will also be positive, the overall expression y = x^2 * (√(3 - x))^2 will be positive too. This means the graph will be above the x-axis for x < 3.
However, when x = 3, the factor (3 - x) becomes zero, which makes the overall expression y = x^2 * (√(3 - x))^2 also zero. This is why the origin is an extreme point on the graph.
So to summarize, the graph lies below the x-axis only when x > 3 because of the factors of the factored form. And since the graph is below the x-axis for x > 3, the origin becomes an extreme point where the graph turns from below to above the x-axis.