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# Step-by-Step Math Answer

Assign variables to the unknowns.

Let (
*
x
*
,
*
y
*
) be a point on the graph of
*
f
*
(
*
x
*
) and let
*
d
*
denote the distance between (
*
x
*
,
*
y
*
) and (4, 0).

Find the primary equation.

The primary equation is

Find the secondary equation.

The secondary equation is

Express the primary equation as a function of
*
x
*
.

Expressing the primary equation in terms of
*
x
*
, we have

where 0
*
x
*
.

The value of
*
x
*
that minimizes
*
d
*
is the same value of
*
x
*
that minimizes the function under the square root.

Let
*
g
*
(
*
x
*
) =
*
x
*
^{
2
}
– 7
*
x
*
+ 16. Then the value of
*
x
*
that minimizes
*
d
*
is the same value of
*
x
*
that minimizes
*
g
*
.

Differentiate
*
g
*
.

Differentiating
*
g
*
, we find that
*
g
*
′ (
*
x
*
) = 2
*
x
*
– 7.

Find the critical number.

Solving the equation
*
g
*
′ (
*
x
*
) = 0, we find that the critical number is
*
x
*
= 7/2.

What should we do next?

Find
*
y
*
.

Verify that
*
g
*
is minimized for this value of
*
x
*
.

Let's verify that
*
g
*
is minimized for this value of
*
x
*
. Begin by finding the second derivative. Differentiating again, we find that
*
g
*
″ (
*
x
*
) = 2.

Apply the Second Derivative Test.

Since
*
g
*
″ (7/2)
0,
*
g
*
is minimized when
*
x
*
= 7/2. Therefore,
*
d
*
is a minimum when
*
x
*
= 7/2.

Use the secondary equation to find
*
y
*
.

Using the secondary equation, we find that

Therefore,
*
d
*
is at a minimum at the point