Chapter 8.3, Problem 21E

To determine

To find:Whether the function has a minimum value or a maximum value.

Answer to Problem 21E

The given function has minimum value −13.

Explanation of Solution

Given function: $y=3x^2-18x+15$

Calculation:

Since the coefficient of $x^2$ is positive, therefore, the given function has minimum value and no maximum value.

To find the minimum value of the function, complete the square as follows.

  $\begin{array}{l}y=3x^2-18x+15\\ y=3\left(x^2-6x\right)+15\\ y=3\left(x^2-6x+9\right)+15-3\times 9\\ y=3{\left(x-3\right)}^2-13\end{array}$

Now,

  $\begin{array}{l}{\left(x-3\right)}^2\ge 0x\in R\\ 3{\left(x-3\right)}^2\ge 0x\in R\\ 3{\left(x-3\right)}^2-13\ge -13x\in R\end{array}$

Therefore, the given function has minimum value −13.