$\"\"$

\

(a) Find the inflection points of $\"\"$ .

\

The function is $\"\"$ .

\

Apply derivative on each side with respect to $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

Again apply derivative on each side with respect to $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

Therefore, inflection points of the graph $\"\"$ is $\"\"$ and $\"\"$ .

\

$\"\"$

\

(b) Determine the equation of the line that intersects both the points.

\

Slope of the two points is $\"\"$ .

\

Substitute $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

Substitute $\"\"$ and $\"\"$ in point-slope form: $\"\"$ .

\

$\"\"$

\

Therefore, the equation of line that intersects both the points is $\"\"$ .

\

$\"\"$

\

(c)

\

Calculate the area bounded by the three regions between the graph $\"\"$ and the line $\"\"$ :

\

Find the intersection points, by equate the function and line equation.

\

$\"\"$

\

Substitute $\"\"$ .

\

$\"\"$

\

$\"\"$ and $\"\"$ .

\

Substitute $\"\"$ .

\

$\"\"$

\

Therefore, the roots are $\"\"$ and $\"\"$ .

\

$\"\"$ .

\

Graph the function $\"\"$ and line equation $\"\"$ .

\

Graph:

\

$\"\"$

\

$\"\"$

\

Area of the region is $\"\"$ .

\

Where $\"\"$ and $\"\"$ .

\

Consider $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Consider $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$ .

\

Consider $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$ .

\

The area between the two inflection points is the sum of the area between the other two regions.

\

$\"\"$

\

(a) The inflection points are $\"\"$ and $\"\"$ .

\

(b) The equation of the line intersects inflection points is $\"\"$ .

\

(c) $\"\"$ , $\"\"$ and $\"\"$ .

\

The area between the two inflection points is the sum of the area between the other two regions.