Chapter 2.1, Problem 53E

### Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Chapter
Section

### Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

# Suppose the graph of revenue as a function of unit price is a parabola that is concave down. What is the significance of the coordinates of the vertex, the x-intercepts, and the y-intercept?

To determine

The significance of the coordinates of the vertex, the x -intercept and the

y -intercept if the graph of revenue as a function of price whose graph of parabola is concave down.

Explanation

Given Information:

The graph of the revenue as a function of price is a parabola with downward facing.

For example:

Consider the Revenue function,

R(p)=ā4p2+100p

Compare the equation R(p)=ā4p2+100p with the standard function f(x)=ax2+bx+c and find the value of a,b and c.

The values are a=ā4,b=100 and c=0.

Here a<0, therefore the parabola will be downward facing and vertex will be the highest point.

Vertex: The formula of p- coordinate or x -coordinate of vertex is,

p=āb2a

Substitute a=ā4 and b=100 in the equation p=āb2a:

p=ā(100)2(ā4)=1008=12.5

This value of p gives the highest point on the graph and thus gives the highest value of revenue R(p).

The R- coordinate or y -coordinate of vertex is,

Substitute p=12.5 in the function R=ā4p2+100p:

R(p)=ā4(12.5)2+100(12.5)=ā4(156.25)+1250=ā625+1250=625

Thus y -coordinate gives the maximum value of the revenue.

p-intercept or x - intercept: To calculate p-intercept of the function, substitute R(p)=0 in the equation R=ā4p2+100p and solve

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