Honeycomb The surface area of a cell in a honeycomb is

In Exercises 1 and 2, decide whether each labeled point is an absolute maximum or minimum, a relative maximum or minimum, or neither y2x

In Exercises 1 and 2, decide whether each labeled point is an absolute maximum or minimum, a relative maximum or minimum, or neither 2x

In Exercises 38, find the value of the derivative (if it exists) at each indicated extremum. 2xx

In Exercises 38, find the value of the derivative (if it exists) at each indicated extremum. xx2

In Exercises 38, find the value of the derivative (if it exists) at each indicated extremum. x2

In Exercises 38, find the value of the derivative (if it exists) at each indicated extremum. x21

In Exercises 38, find the value of the derivative (if it exists) at each indicated extremum. x21x

In Exercises 38, find the value of the derivative (if it exists) at each indicated extremum. 21x2

In Exercises 912, approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, relative minimum, absolute maximum, absolute minimum, or none of these at each critical number on the interval shown. 1x2y

In Exercises 912, approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, relative minimum, absolute maximum, absolute minimum, or none of these at each critical number on the interval shown. 1x2y

In Exercises 912, approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, relative minimum, absolute maximum, absolute minimum, or none of these at each critical number on the interval shown. x2yx

In Exercises 912, approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, relative minimum, absolute maximum, absolute minimum, or none of these at each critical number on the interval shown. 2yx2

In Exercises 1318, find any critical numbers of the function. yx2

In Exercises 1318, find any critical numbers of the function. x24

In Exercises 1318, find any critical numbers of the function. x24y

In Exercises 1318, find any critical numbers of the function. 24y

In Exercises 1318, find any critical numbers of the function. 4y0

In Exercises 1318, find any critical numbers of the function. 4y0y

In Exercises 1936, locate the absolute extrema of the function on the closed interval. y0y

In Exercises 1936, locate the absolute extrema of the function on the closed interval. 0yx

In Exercises 1936, locate the absolute extrema of the function on the closed interval. 0yx2

In Exercises 1936, locate the absolute extrema of the function on the closed interval. yx2

In Exercises 1936, locate the absolute extrema of the function on the closed interval. x23

In Exercises 1936, locate the absolute extrema of the function on the closed interval. x23x

In Exercises 1936, locate the absolute extrema of the function on the closed interval. 23x

In Exercises 1936, locate the absolute extrema of the function on the closed interval. 3x3

In Exercises 1936, locate the absolute extrema of the function on the closed interval. 3x3x

In Exercises 1936, locate the absolute extrema of the function on the closed interval. x3x

In Exercises 1936, locate the absolute extrema of the function on the closed interval. 3x1

In Exercises 1936, locate the absolute extrema of the function on the closed interval. 3x1

In Exercises 1936, locate the absolute extrema of the function on the closed interval. x12

In Exercises 1936, locate the absolute extrema of the function on the closed interval. 12y

In Exercises 1936, locate the absolute extrema of the function on the closed interval. 12y

In Exercises 1936, locate the absolute extrema of the function on the closed interval. 2y3

In Exercises 1936, locate the absolute extrema of the function on the closed interval. 2y3

In Exercises 1936, locate the absolute extrema of the function on the closed interval. y32

In Exercises 3740, locate the absolute extrema of the function (if any exist) over each interval. 32

In Exercises 3740, locate the absolute extrema of the function (if any exist) over each interval. 32

In Exercises 3740, locate the absolute extrema of the function (if any exist) over each interval. 2x

In Exercises 3740, locate the absolute extrema of the function (if any exist) over each interval. 2x

In Exercises 4144, sketch the graph of the function. Then locate the absolute extrema of the function over the given interval. xx

In Exercises 4144, sketch the graph of the function. Then locate the absolute extrema of the function over the given interval. xxy

In Exercises 4144, sketch the graph of the function. Then locate the absolute extrema of the function over the given interval. xxy

In Exercises 4144, sketch the graph of the function. Then locate the absolute extrema of the function over the given interval. xy

In Exercises 45 and 46, use a graphing utility to graph the function. Then locate the absolute extrema of the function over the given interval. xyx

In Exercises 45 and 46, use a graphing utility to graph the function. Then locate the absolute extrema of the function over the given interval. yx

In Exercises 47 and 48, (a) use a computer algebra system to graph the function and approximate any absolute extrema on the given interval. (b) Use the utility to find any critical numbers, and use them to find any absolute extrema not located at the endpoints. Compare the results with those in part (a). x1

In Exercises 47 and 48, (a) use a computer algebra system to graph the function and approximate any absolute extrema on the given interval. (b) Use the utility to find any critical numbers, and use them to find any absolute extrema not located at the endpoints. Compare the results with those in part (a). x1

In Exercises 49 and 50, use a computer algebra system to find the maximum value of on the closed interval. (This value is used in the error estimate for the Trapezoidal Rule, as discussed in Section 4.6.) x1

In Exercises 49 and 50, use a computer algebra system to find the maximum value of on the closed interval. (This value is used in the error estimate for the Trapezoidal Rule, as discussed in Section 4.6.) 1x

In Exercises 51 and 52, use a computer algebra system to find the maximum value of on the closed interval. (This value is used in the error estimate for Simpsons Rule, as discussed in Section 4.6.) 1x2

In Exercises 51 and 52, use a computer algebra system to find the maximum value of on the closed interval. (This value is used in the error estimate for Simpsons Rule, as discussed in Section 4.6.) x2

In Exercises 53 and 54, graph a function on the interval having the given characteristics. Absolute maximum at , absolute minimum at , relative maximum at x21

In Exercises 53 and 54, graph a function on the interval having the given characteristics. Relative minimum at , critical number at but no extrema, absolute maximum at , absolute minimum at x21y

In Exercises 5558, determine from the graph whether has a minimum in the open interval 21y

In Exercises 5558, determine from the graph whether has a minimum in the open interval 1yx

In Exercises 5558, determine from the graph whether has a minimum in the open interval 1yx2

In Exercises 5558, determine from the graph whether has a minimum in the open interval yx2

Power The formula for the power output of a battery is where is the electromotive force in volts, is the resistance, and is the current. Find the current (measured in amperes) that corresponds to a maximum value of in a battery for which volts and ohm. Assume that a 15-ampere fuse bounds the output in the interval Could the power output be increased by replacing the 15-ampere fuse with a 20-ampere fuse? Explain. x225

Inventory Cost A retailer has determined that the cost of ordering and storing units of a product is The delivery truck can bring at most 300 units per order. Find the order size that will minimize cost. Could the cost be decreased if the truck were replaced with one that could bring at most 400 units? Explain. x225

Lawn Sprinkler A lawn sprinkler is constructed in such a way that is constant, where ranges between and (see figure). The distance the water travels horizontally is where is the speed of the water. Find and explain why this lawn sprinkler does not water evenly. What part of the lawn receives the most water? 225x

Honeycomb The surface area of a cell in a honeycomb is where and are positive constants and is the angle at which the upper faces meet the altitude of the cell (see figure). Find the angle that minimizes the surface area 25x2

True or False? In Exercises 6366, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The maximum of a function that is continuous on a closed interval can occur at two different values in the interval. 25x2y

True or False? In Exercises 6366, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function is continuous on a closed interval, then it must have a minimum on the interval. x2y2

True or False? In Exercises 6366, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If is a critical number of the function then it is also a critical number of the function where is a constant. x2y2

True or False? In Exercises 6366, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If is a critical number of the function then it is also a critical number of the function where is a constant. 2y2x

Let the function be differentiable on an interval containing If has a maximum value at show that has a minimum value at y2x3

Consider the cubic function where Show that can have zero, one, or two critical numbers and give an example of each case. 2x3

Highway Design In order to build a highway, it is necessary to fill a section of a valley where the grades (slopes) of the sides are 9% and 6% (see figure). The top of the filled region will have the shape of a parabolic arc that is tangent to the two slopes at the points and The horizontal distance between the points and is 1000 feet. (a) Find a quadratic function that describes the top of the filled region. (b) Construct a table giving the depths of the fill for 0, 100, 200, 300, 400, and 500. (c) What will be the lowest point on the completed highway? Will it be directly over the point where the two hillsides come together? x34