Chapter 4.2, Problem 62SR

To determine

To check: whether the given function has maximum or minimum value, and find the value. Also, state the domain and range of the function.

Answer to Problem 62SR

Minimum value $y=-30$ ; and domain is $\left\{f\left(x\right)|f\left(x\right)>-30\right\}$ .

Explanation of Solution

Given:

The given expression is $f\left(x\right)=3x^2+12x-18$ .

Calculation:

Consider the following expression,

  $f\left(x\right)=3x^2+12x-18$

For the above function,

  $a=3$

As, the graph opens upwards the function has a minimum value.

The coordinate of the vertex will be,

  $\begin{array}{l}x=\frac{-b}{2a}\\ x=\frac{-12}{2\left(3\right)}\{\text{Substitute these values}}\\ x=-2\end{array}$

So, the y- coordinate will be,

Plug the value for x in the above coordinate,

  $\begin{array}{l}y=3x^2+12x-18\\ y=3{\left(-2\right)}^2+12\left(-2\right)-18\\ y=-30\end{array}$

So, the maximum value of the function is $y=-30$ .

The domain of the above function is all real numbers.

The range is all the real numbers less than or equal to $-30$ .

Or, $\left\{f\left(x\right)|f\left(x\right)>-30\right\}$ .

Hence, the given function has maximum value and the value is $y=-30$ .

And the domain belongs to R, and the range is $\left\{f\left(x\right)|f\left(x\right)>-30\right\}$ .