Question

The profit from a business is

described by the function
P(x) = -5x² + 30x + 8, where x
is the number of items made,
in thousands, and P(x) is the
profit in dollars. How many
items will maximize the profit?

Answer 1

First of all we will understand the question!!

The question is saying that you are given a function and you have to find the value of x which will give the maximum profit... Lets solve it by finding the extrema using the vertex

`\rm \: p(x) = - 5 {x}^(2) + 30x + 8`

• Identify the coefficients a and b of the quadratic function

`\rm \: p(x) = { - 5x}^(2) + 30x + 8 \\ \rm \: a = - 5 \: and \: b \: = 30`

• Since a<0, the function has the maximum value at x, calculated by substituting a and b into x=-b/2a

`\rm \: x = (30)/( 2 * (- 5))`

• Solve the equation for x

`\rm \: x = 3`

• The maximum of the quadratic function is at x=3