This cubic, c(x) = x^3 – 2x^2 – 24x, has zeroes at x ∈ {-4, 0, 6}. What is the approximate value of its local minimum? @ As an approximation, we treat cubics, and other higher-order polynomials, as quadratics between their zeroes. Then, recognizing that the local minimum is between the second two zeroes, the local minimum is found at the midpoint between these zeroes.
a. -63
b. 21
c. 0
d. no way to know this
We draw the graph
The local minimum looks to be where the litle plus sign is:
We can't find it exactly with algebra (although it can be with
calculus, as you will learn if you take calculus), but in
algebra, we can approximate it by assuming the curve is like
the green parabola below. We can see it's not exactly the same,
but it will come fairly close.
We know that the x-coordinate of the vertex of the green parabola
is half-way between its x-intercepts at 0 and 6.
Halfway between the x-intercepts (0,0,) and (6,0) is the point
(3,0), so we will substitute x=3 into the equation
So the approximation of the local minimum, which is the vertex of
the closely fitting green parabola, (3,-63), which is the second
point marked below.
So the correct answer is a. -63.
As you see the two points are not the same, but that's as good as
we can do with just algebra, and no calculus.
Edwin