Chapter 1.1, Problem 2E

Solution

To describe: The determination of maximum or minimum value of a quadratic function.

The minima and maxima of the quadratic function can be determined from the coordinates of the vertex of parabola represented by the quadric function.

Given data: A quadratic function has minimum or maximum value.

Explanation:

A quadratic function represents a parabola and minima or maxima of the quadratic function exists the vertex of that parabola.

The minima and maxima of the quadratic function can be determined from the coordinates of the vertex of parabola represented by the quadric function.

Let $f\left(x\right)=a{x}^2+bx+c$ is a parabola. The vertex of the parabola represented by this function is $\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right)$ .

Illustration-1: Consider quadratic function is $f\left(x\right)=2x^2+5x+8$ , the vertex of the parabola represented by function is $\left(-\frac{5}{2\cdot 2},f\left(-\frac{5}{2\cdot 2}\right)\right)=\left(-\frac{5}{4},f\left(-\frac{5}{4}\right)\right)$ . Now, $f\left(-\frac{5}{4}\right)$ is

  $\begin{array}{c}f\left(x\right)=2{\left(-\frac{5}{4}\right)}^2+5\left(-\frac{5}{4}\right)+8\\ =\frac{25}{8}-\frac{25}{2}+8\\ =4.875\end{array}$ \

If coefficient of $x^2$ is positive, then $f\left(-\frac{b}{2a}\right)$ is the minimum value of the function and if coefficient of $x^2$ is negative, then $f\left(-\frac{b}{2a}\right)$ is the maximum value of the function that is evident from the graphs of the functions, $f\left(x\right)=2x^2+5x+8$ and $g\left(x\right)=-2x^2+5x+8$ .