Mathematics

Problem

Consider the one-parameter family of functions defined by

y = y_0 + \int_{2}^{x} \bigl(3t^{2} - 6t + 4\bigr)\,dt,

where

y_0

is a real constant. Each member of this family satisfies the differential equation

\frac{dy}{dx} = 3x^{2} - 6x + 4

and passes through the point

(2, y_0)

. a. Find an explicit polynomial expression

F_{y_0}(x)

for the function

y

in terms of

x

and

y_0

. b. Describe precisely how changing the value of

y_0

affects the graph of

F_{y_0}(x)

in the

xy

-plane. c. Determine the value of

y_0

for which the corresponding curve passes through the point

(4,10)

. Show your reasoning. d. Give a mathematical argument to explain why no two distinct values of

y_0

can produce curves that intersect each other at any value of

x