1. ## turning points

A publisher has to choose a price £x for a new book. The total amount of money she will receive from sales is £y, where y = 20000x - 5000xsquared

a) find the price which gives the maximum amount of money from sales?

Is it f'(x) = 20000 - 10000x

So when the book cost £0 will she receive the maximum amount of money which is £20000.

This does not much sense?

b) find the maximum amount of money from sales.

£20000

2. Solve f'(x)=0

what do you get?

3. Originally Posted by Natasha1
A publisher has to choose a price £x for a new book. The total amount of money she will receive from sales is £y, where y = 20000x - 5000xsquared

a) find the price which gives the maximum amount of money from sales?

Is it f'(x) = 20000 - 10000x

So when the book cost £0 will she receive the maximum amount of money which is £20000.

No, if you think in terms of the graph, it's clear

This does not much sense? exactly

b) find the maximum amount of money from sales.

£20000
Hi Natasha,

Do you know what the graphs of quadratic equations look like ?

$y=-5000x^2+20,000x$ is a parabola with a maximum value.

It's like an upside-down U-shape because the multiplier of the highest power of x is negative.

You should study the graphs of quadratic equations to see what they look like.

For this type of graph, the tangent on the top of the curve, at the maximum value , is horizontal.
A horizontal line has a slope of zero.
You get a formula for the slope of the tangent by calculating the derivative.

Hence, at the maximum, what equation should you have for the derivative ?

4. f'(x) = 20000 - 10000x = 0
so x = 2

hence substituting back into y = 20000x - 5000xsquared we get

y = 40000 - 20000
y = 20000