# Functions, Zeros And Application

Given f(x) = ex2 + fx + g, where e, f, and g are real numbers,

(a) Find the zeros of the function in terms of e, f, and g.

(b) Under what condition would the zeros of the given function be two distinct real values?

(c) Determine the x-coordinate (the x-value) of the point that would represent the minimum or the maximum of the graph of the function by taking the average (mean value) of the zeros of the function found in Part (a).

(d) Under what condition would the graph of the function have a maximum, not a minimum, with the x-value obtained in Part (c)?

(e) Draw the graph of a function, with numerical coefficients of your choice, to represent your thought on what you stated in Part (d).

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

I have copied and pasted the content of this file. Hence it will not show certain formula and/or the graphs. Please see attachment for complete solution.

(a) f(x) = ex2 + fx + g

f(x) = ex2 + fx + g = (x - x1) (x-x2) where x1 and x2 are the solution to the quadratic formula f(x)

Using the quadratic formula the solution to f(x) are,

x = [-f Â± âˆš(f2 - 4eg)]/2e

Hence x1 = -f + âˆš(f2 - 4eg)/2e and x2 = -f - âˆš(f2 - 4eg)/2e

f(x) = (x - x1) (x-x2)

zeros of f(x) are x1 and x2

zeros of f(x) are -f + âˆš(f2 - 4eg)/2e and -f - âˆš(f2 - 4eg)/2e

(b) The two zeros are distinct when the term inside the square-root is not zero and positive.

(note when it is negative we would not have real zero)

The condition is therefore, f2 - 4eg > 0

(c) To find the max or min of the graph, we need to find the first and second derivative of f(x).

Average of the zeros found in part(a) is x1+ x2)/2 = -2f/2e / 2 = -f/2e

This represents the coordinate of the max or min of the function.

(d) For maximum the condition is that the second derivative of the function is less than zero.

f' = 2ex + f

f" = 2e < 0

e < 0

(e) Let e = -1 (so we should have a maximum)

Let f = 1 and g = 2 (so f2 - 4eg > 0 and we should have two distinct zeros)

f(x) = -x2 + x + 2

Graph is in the next page. It shows a maximum and two zeros.

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