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question06

Page history last edited by PBworks 16 years, 5 months ago

Question 6

 

The use of a graphing calculator is NOT allowed.

 

The figure shows ƒ', the derivative of a function ƒ.

The domain of the function ƒ is the interval [0, 20].

 

(a) For what values of x, 0 < x < 20, does ƒ have a relative maximum? Justify your answer.

 

(b) For what values of x is the graph of ƒ concave down? Justify your answer.

 

(c) If ƒ(0) = 10, sketch a graph of the function ƒ on axes similar to the ones provided below. List the coordinates of all critical points and inflection points.

 

 

 

Solution

 

The relative maximum depends on the where the graph of the derivative of f equals zero. Due to the fact that the function f, exists from 0 to 20 including both endpoints. Between 0 and 20, the maximum will be visible on f' because it will be above the x axis before it crosses the x axis and exists below the x axis.

There are three x coordinates that need to be considered for this problem. The obvious x=10, and the endpoints, x=0 and x=20. No one knows if x=10 is greater than x=0, but if you take the integral from 0 to 10, you will find that out.

 

Σ(this takes the place of the integral sign)

Σ(0,10) Can be found by finding the area of a triangle.

A = b(h)/2

A = 10(10)/2

A = 50

Because A between 0 and 10 is positive,when x=10 the value of f is greater than when x=0.

Now that x=0 has been eliminated as a choice for maximum, the other contender needs to be compared to x=10.

Take the integral from 10 to 20, and if it is positive, then x=20 is the max, if negative, x=10 is the max.

Σ(10,15) is a triangle.

A = b(h)/2

A = 5(-5)/2

A = -12.5

Σ(15,20) is also a triangle.

A = b(h)/2

A = 5(10)/2

A = 25

Σ(10,20)(f'(x))= 25-12.5

Σ(10,20)(f'(x))= 12.5

Because the integral from 10 to 20 is positive, the relative maximum occurs when x=20.

Answer to Part a) The relative maximum occurs when x=20.

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