Chapter 7.4, Problem 27E

Solution

The system of inequality, whose solution is the region shaded in pink in the given figure and all the boundaries are to be included.

The system of inequalities is $y\ge -x^2+1$ and $x^2+y^2\le 4$ .

Given Information:

The figure is show as,

  

Explanation:

Consider the given system of equations,

  

It can be observed that in the given figure, one figure is circle. The general form of the circle is ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ . The circle is cantered at origin and the radius is 2. So,

  $\begin{array}{c}{\left(x-0\right)}^2+{\left(y-0\right)}^2=2^2\\ x^2+y^2=4\end{array}$

Using the test point as origin, which is inside the circle and the appropriate inequality should be used $\le$ . So,

  $x^2+y^2\le 4$

It can be observed that in the given figure, second figure is parabola with downward. The general form of the circle is $y=a{\left(x-h\right)}^2+k$ . The parabola vertex is $\left(0,1\right)$ , which is the minimum value and the intercepts is $\left(\pm 1,0\right)$ .

  $y=a{\left(x-h\right)}^2+k$

So, substitute all the values in the equation to find the value of a .

  $\begin{array}{c}0=a{\left(1-0\right)}^2+1\\ -1=a\cdot 1\\ a=-1\end{array}$

Thus, the obtained equation is $y=-x^2+1$ .

Using the test point as origin, which is below the parabola and the appropriate inequality should be used $\ge$ . Then,

  $y\ge -x^2+1$

Hence, the obtained system of inequalities are $y\ge -x^2+1$ and $x^2+y^2\le 4$ .