Chapter 5.7, Problem 6E

To determine

Toidentify:The real zeros, maximum and minimum points, number and type of zeros, y-intercept, and symmetry of the given function by graphing the function.

Answer to Problem 6E

The real zeros of function $k\left(x\right)$ are $x=\pm 1.732$ and $x=\pm 2.236$ . The maximum and minimum points of the function $k\left(x\right)$ are $\left(0,15\right)$ and $\left(\pm 2,-1\right)$ respectively. There are total 4 real zeros of function $k\left(x\right)$ and each with multiplicity 1. The y-intercept of the function is $\left(0,15\right)$ . The line $x=0$ is the axis of symmetry of the function.

Explanation of Solution

Given information:

The function is $k\left(x\right)=x^4-8x^2+15$ .

Method used:

The graph of function is drawn using graphing calculator.

Graph and calculations:

The graph of the function $k\left(x\right)=x^4-8x^2+15$ is shown in Figure-1 here.

Zeros of$k\left(x\right)$:

From the graph:

The real zeros of function $k\left(x\right)$ are: $x=\pm 1.732$ and $x=\pm 2.236$ each with multiplicity 1.

Maximum and Minimum points of$k\left(x\right)$ :

The maximum point of $k\left(x\right)$ is $\left(0,15\right)$ and minimum points $\left(\pm 2,-1\right)$ .

Number and type of zeros of $k\left(x\right)$

There are total 4real zeros of function $k\left(x\right)$ each of multiplicity 1.

They-intercept $k\left(x\right)$

The function intersectsy−axis at point $\left(0,15\right)$ that is the y-intercept of $k\left(x\right)$ .

Symmetry of $k\left(x\right)$

The graph is symmetrical about y-axis, hence, line $x=0$ is the axis of symmetry.

Conclusion:

The real zeros of function $k\left(x\right)$ are $x=\pm 1.732$ and $x=\pm 2.236$ .The maximum and minimum points of the function $k\left(x\right)$ are $\left(0,15\right)$ and $\left(\pm 2,-1\right)$ respectively. There are total 4 real zeros of function $k\left(x\right)$ and each with multiplicity 1. The y-intercept of the function is $\left(0,15\right)$ . The line $x=0$ is the axis of symmetry of the function.