Q. 67 - Calculus

Question

Find the intervals on which each function in Exercises 67–74 is positive or negative. Make clear how your work uses the Intermediate Value Theorem and continuity. You may assume that polynomials and their quotients are continuous on the intervals on which they are defined.

$f\left(x\right)=2+5x+2x^2$

Step-by-Step Solution

Answer

The intervals on which the function $f\left(x\right)=2+5x+2x^2$ is positive is $[-\infty ,-2]\cup [-\frac{1}{2},\infty ]$ and negative on $[-2,-\frac{1}{2}]$

Step 1. Given Information.

The function:

$f\left(x\right)=2+5x+2x^2$

Step 2. Find roots of the function

$$\begin{gathered} f\left(x\right)=2+5x+2x^2 \\ =2x^2+4x+x+2 \\ =(2x+1)(x+2) \\ x=-\frac{1}{2},-2 \end{gathered}$$

Step 3. Evaluate at some points.

By theorem "A function f can change sign (from positive to negative or vice versa) at a point x = c only if f(x) is zero, undefined, or discontinuous at x = c."

$f(-3)=5; f(-1)=-1; f\left(2\right)=20$

Step 4. Sketch the graph.

The graph of the function is

From the graph, we can conclude that the function is positive on the interval $[-\infty ,-2]\cup [-\frac{1}{2},\infty ]$ and negative on the interval $[-2,-\frac{1}{2}]$