Question

The following polynomial represents a profit function for a certain production line, where x is a number of produced units, find a. the zeros and the multiplicity of each b. where the graph crosses or touches the x‐axis c. number of turning points d. explain the meaning of the above, if any f(x)=(x+5)(x-3)(x+3)

Answer 1

Graph crosses the x axis at y = 0
y= 0 when either x+5 = 0
x-3 = 0
and/or x+3 = 0
(as anything times 0 = 0)
so from that you can get your turning points:
x+5 = 0, therefore x=-5
x-3 = 0, therefore x=3
x+3 = 0, therefore x=-3
your function crosses the x axis at the points x= -5, -3, and 3
Do you want me to move on to the next part?

Answer 2

So, as you can see, it's a cubic polynomial, with 2 turning points
to work them out mathematically, multiply out the brackets so you're left with the form:
f(x) = ax^3 + bx^2 + cx +d

in this case:
f(x) = (x+5)(x−3)(x+3)
f(x) = x^3 + 5x^2 -9x -45

Answer 3

so far so good, now how do I explain (d)

Answer 4

Now, to find the turning points:
they will occur when the gradient of the graph = 0
the equation for gradient is the derivative of the equation of the function

so, the equation for your function:

f(x) = x^3 + 5x^2 -9x -45

therefore derivative of function:

f '(x) = 3x^2 + 10x -9m


Now, you can use the quadratic formula to find the exact position of when

3x^2 + 10x -9 = 0, which is the x co-ordinates of the turning points

In this case it's at:
x= -4.07 ish and
x= 0.737 ish

so as it's a profitability graph, and x = number of units produced... i'd assume that y = profits made...? maybe...?
i'm not sure how to do the explanation part, sorry, just the maths

Answer 5

that's cool, thank you so very much I appreciate your help.

Answer 6

Your Welcome!!:)