Maths for Engineers and Scientists 1

Maths for Engineers and Scientists 2

Optimization: Stationary Points, Maxima/Minima and Modeling

• The point c is called a stationary point, if
• Second derivative test for local maxima and minima:
• if and then is a local maximum.
• • if and then is a local minimum.
• • if and then use method from previous course.

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Global and local maxima and minima

• At a point , the function has
• a local maximum if for all near
• a local minimum if for all near
• and the function has
• a global maximum if for all allowable values of
• a global minimum if for all allowable values of .

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If is a continuous, smooth function defined for the allowable range of values then global maxima and minima exist. To find them you need to check the values at

• If is a continuous, smooth function defined for the allowable range of values then global maxima and minima exist. To find them you need to check the values at
• all the local extrema
• the ends of the interval (i.e. and )

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Example 1. Locate the local and global extrema in the figure above.

• Example 1. Locate the local and global extrema in the figure above.

Example 2. Locate the extrema in the plots below.

• Example 2. Locate the extrema in the plots below.

Modeling and optimization examples

• Example 3. A rectangle is to be made with a perimeter of 16m. Find the lengths of the
• sides when the area is a maximum.
• Example 4. An open rectangular tank with a square base is to be constructed so that it has a volume of 500 cubic meters. Find the dimensions of the tank which minimize its total surface area (and hence the amount of material used).

Example 5. A rectangular metal plate has squares cut from its four

• Example 5. A rectangular metal plate has squares cut from its four
• corners as illustrated and is then folded up to make an open box. Why do we need the restriction ? What is the dimension of this box that has the maximum volume?

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Example 6. A closed rectangular box has a height and width . Its length is twice its width. It has a fixed outer surface area of . Find the value of and required to make the volume of the box a maximum. Calculate the maximum volume.

• Example 6. A closed rectangular box has a height and width . Its length is twice its width. It has a fixed outer surface area of . Find the value of and required to make the volume of the box a maximum. Calculate the maximum volume.

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Example 7.A rectangular plot of farmland is enclosed by of fencing material on three sides. The fourth side of the plot is bordered by a stone wall .Find the dimensions of the plot that enclose the maximum area. Find the maximum area.

• Example 7.A rectangular plot of farmland is enclosed by of fencing material on three sides. The fourth side of the plot is bordered by a stone wall .Find the dimensions of the plot that enclose the maximum area. Find the maximum area.
• Example 8. Find the dimensions of an open box with a square base and surface area of 192 that has a maximum volume.

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Example 9. A cylinder is inscribed in a cone with radius and height . Find the radius and height of cylinder with maximum volume.

• Example 9. A cylinder is inscribed in a cone with radius and height . Find the radius and height of cylinder with maximum volume.

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Example 10. Find the maximum and minimum values of the function

• Example 10. Find the maximum and minimum values of the function
• on the interval .

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Example 11. A cuboid container with a base length twice its width is to be made with of metal.

• Example 11. A cuboid container with a base length twice its width is to be made with of metal.
• Show that the height , where is the width of the base.
• Express the volume, , in terms of .
• Find the maximum volume.

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