Chapter 7.6, Problem 5OE

To determine

Whether the function $f\left(x\right)={\left(x-5\right)}^2$ has a maximum or minimum value and then give the value of $\left(x\right)$ associated with it.

Answer to Problem 5OE

The function $f\left(x\right)={\left(x-5\right)}^2$ will have minimum value and $x=5$ is the associated value of $\left(x\right)$ which will give the minimum value of function equal to $0$ .

Explanation of Solution

Given information: The given function is $f\left(x\right)={\left(x-5\right)}^2$ which can also be written as,

  $f\left(x\right)=x^2-10x+25\mathrm{....}\left(1\right)$

Formula used: The function of the form $f\left(x\right)=a{x}^2+bx+c$ where $a\ne 0$ will have the maximum value if $a<0$ and minimum value when $a>0$ . Mathematically,

  $\left(\begin{array}{l}f\left(x\right)=a{x}^2+bx+c\text{ is maximum for }a<0\\ \text{ is minimum for }a>0\end{array}\right)\mathrm{....}\left(2\right)$

Calculation: Comparing the formula (2) with (1) gives, $a=1$ which is greater than zero. Therefore the given function $f\left(x\right)={\left(x-5\right)}^2$ will give the minimum value.

Now the value for $\left(x\right)$ associated with minimum value can be evaluated from the formula $x=-\frac{b}{2a}$ from (1),

  $\begin{array}{c}x=-\frac{\left(-10\right)}{2\left(1\right)}\mathrm{....}\left(a=1,b=-10\right)\\ x=5\end{array}$

Hence the value of function will give the minimum value at $x=5$ . Now evaluate the minimum value at $x=5$ from (1).

  $\begin{array}{c}{\left[f\left(5\right)\right]}_{\min }={\left(5\right)}^2-10\left(5\right)+25\mathrm{....}\left(1\right)\\ =0\end{array}$