Chapter 1, Problem 30T

Solution

Graph the system of inequalities consisting of $y<x^2\text{ and }y\ge \frac{1}{2}{\left(x+1\right)}^2$ .

Given:

The inequalities are $y<x^2\text{ and }y\ge \frac{1}{2}{\left(x+1\right)}^2$ .

Graph:

The graph for the system of inequalities $y<x^2\text{ and }y\ge \frac{1}{2}{\left(x+1\right)}^2$ is shown below.

  

Interpretation:

From the graph it can be interpreted that the graph for $y<x^2$ is an upward parabola with the dotted parabolic curve in green and its region lies outside of it and the graph for $y\ge \frac{1}{2}{\left(x+1\right)}^2$ is also an upward parabola with the bold parabolic curve. Therefore, the common region representing the system of inequalities $y<x^2\text{ and }y\ge \frac{1}{2}{\left(x+1\right)}^2$ is shown in graph.