Chapter 11, Problem 21STP

To determine

To evaluate: the value of ‘ x’ that maximizes the profit earned per item.

Answer to Problem 21STP

The value of x that maximizes the profit earned per item is $\frac{3}{2}$ dollars.

Explanation of Solution

Given:

The function

  $p\left(x\right)=-4x^2+12x-5$

represents the profit earned (in dollars) for an item that costs x dollars.

Calculation: -

Finding the first derivative,

  $\begin{array}{l}p\left(x\right)=-4x^2+12x-5\\ p\text{'}\left(x\right)=-8x+12\end{array}$

Putting first derivative equals to 0,

  $\begin{array}{c}-8x+12=0\\ x=\frac{12}{8}\\ x=\frac{3}{2}\end{array}$

So,

The value of ‘ x’ that maximizes the profit earned per item is $\frac{3}{2}$ dollars.