Chapter 10.9, Problem 30E

To determine

To find: the coordinates of all the vertices and determine whether the solution set bounds or not.

Answer to Problem 30E

The vertices are $[(0,0),(0,4)\text{ and }(2,4)$

The solution set is bounded

Explanation of Solution

Given:

  $\begin{array}{l}y\ge x^2,\\ y\le 4,\\ x\ge 0\end{array}$

Calculation:

We obtain the equations of the parabola and the straight line from the given inequalities:

  $y=x^2$

  $y=4$

Solving these two equations we get the intersection point of these two curves.

Substitute $y=4$ into the equation $y=x^2$ we have

  $x^2=4$

  $x=\pm 2$

Substitute $x=2$ into $y=x^2$ we get $y=4$

Substitute $x=-2$ into $y=x^2$ we get $y=4$

So, the intersection points are (-2,4) and (2,4) .

Some points on the parabola $y=x^2$ are

X	-3	-2	2	0	3
Y	9	4	4	0	9

Using these points we first draw the graph of this parabola.

We graph the straight line and the parabolas given by the equation that corresponds to the inequality. To determine the graph of the inequality, we need to check our test point. Here, we take (0,1) as test point.

Inequality	Test point (0,1)	Conclusion
$y\ge x^2$	$1\ge 0$	Satisfies inequality
$y\le 4$	$1\le 4$	Satisfies inequality

Since (0,1) is below the line $y=4$ and above the parabola $y=x^2,$ our check shows that the region on or below the line and on or above the parabola must satisfy the inequalities. The inequality $x\ge 0$ shows that $x$ is nonnegative

Therefore, graph with shaded region is

  

Vertices: The co-ordinate of one vertex is the point of intersection of the given line and the parabola in the first quadrant, which is clearly (2,4). The origin (0,0) is also a vertex. The other vertex is the $y$ intercept of the corresponding line: (0,4) .

Therefore, the vertices are $[(0,0),(0,4)\text{ and }(2,4)$

From the figure it is clear that the shaded region is bounded. Therefore, the solution set is bounded

Conclusion:

Therefore, the vertices are $[(0,0),(0,4)\text{ and }(2,4)$ and the solution set is bounded