Derivative functions > Calculating extreme values
1234Calculating extreme values

Exercises

Exercise 1

Here you see the graph of the function f ( x ) = x 4 - 8 x 2 .

Determine the extreme values of this function by differentiating.

Exercise 2

Given the functions f ( x ) = 4000 - 10 x 2 and g ( x ) = ( x - 10 ) ( x 2 - 400 ) .

a

In order to get a good view of both functions on your graphing calculator, you will need to adjust the settings. First you should determine the x-intersects of both functions.

b

Now you know what settings you need to use for x . Next you should determine the extrema of both functions.

c

You can now get a good view of both functions on your graphing calculator. Solve: f ( x ) g ( x ) .

Exercise 3

Around a rectangular sports field there is an athletic track consisting of two stright pieces and two half circles. The total length of the track is 400 m. The dimensions of the field are chosen such that it's surface area is maximised.

What are the dimensions of the field?

Exercise 4

A manufacturer sells self-rising flour at € 2,25 per kilogram. The total costs T C for production and storage are:

q (in hundreds of kg) 1 2 3 4 5 6
TC (in euro) 75 100 125 200 400 800
a

By how much does the profit per kilogram increase when production increases from 400 to 500 kg?

b

The manufacturer was given the following formula for the total costs: T C = 10 q 3 - 60 q 2 + 130 q . Check if this formula fits the numbers in the table above.

c

Set up a formula for the total profit as a function of q .

d

Calculate the marginal gains M G at a production of 400 kilo using M G ( q ) = T P ' ( q ) . What is the economic significance of this value?

e

Calculate the maximum profit using the function M G .

Exercise 5

For any value of a there is a function f ( x ) = x 4 - a x 2 .

a

What values of a result in a minimum of the function at -1 ?

b

A tangent of the graph of f at x = 1 goes through point ( 0 , 4 ) . What is the required value of a ?

Exercise 6

For every positive value of p there is a function with rule f ( x ) = x 3 - 6 p x 2 - 16 .

a

Does every one of these functions have extrema? Provide an explanation for your answer.

b

What value of p results in an extremum with a function value of -32 ? Is this a minimum or a maximum?

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