On the west coast of Canada, crows eat whelks (a shellfish). To open the whelks, the

For Problems 12, indicate all critical points on the given graphs. Determine which correspond to local minima, local maxima, global minima, global maxima, or none of these. (Note that the graphs are on closed intervals.)

For Problems 12, indicate all critical points on the given graphs. Determine which correspond to local minima, local maxima, global minima, global maxima, or none of these. (Note that the graphs are on closed intervals.)

For each interval, use Figure 4.39 to choose the statement that gives the location of the global maximum and global minimum of f on the interval. (a) 4 x 12 (b) 11 x 16 (c) 4 x 9 (d) 8 x 18 (I) Maximum at right endpoint, minimum at left endpoint. (II) Maximum at right endpoint, minimum at critical point. (III) Maximum at left endpoint, minimum at right endpoint. (IV) Maximum at left endpoint, minimum at critical point. 10 20 5 10 15 f(x) x Figure 4.39

Has local minimum and global minimum at x = 3but no local or global maximum.

Has local minimum at x = 3, local maximum at x = 8, but no global maximum or minimum.

Has no local or global maxima or minima.

Has local and global minimum at x = 3, local and global maximum at x = 8.

True or false? Give an explanation for your answer. The global maximum of f(x) = x2 on every closed interval is at one of the endpoints of the interval.

Plot the graph of f(x) = x3ex using a graphing calculator or computer to find all local and global maxima and minima for: (a) 1 x 4 (b) 3 x 2

Has local minimum at x = 3, local maximum at x = 8, but global maximum and global minimum at the endpoints of the interval.

Has local and global maximum at x = 3, local and global minimum at x = 10.

Has global maximum at x = 0, global minimum at x = 10, and no other localmaxima orminima.

Has local and global minimum at x = 3, local and global maximum at x = 8.

The function y = t(x) is positive and continuous with a global maximum at the point (3, 3). Graph t(x) if t(x) and t(x) have the same sign for x < 3, but opposite signs forx > 3.

Figure 4.40 shows the rate at which photosynthesis is taking place in a leaf. (a) At what time, approximately, is photosynthesis proceeding fastest for t 0? (b) If the leaf grows at a rate proportional to the rate of photosynthesis, for what part of the interval 0 t 200 is the leaf growing? When is it growing fastest? 100 200 t (days) rate of photosynthesis (oxygen/time) Figure 4.40

For the functions in Problems 1619, do the following: (a) Find f and f . (b) Find the critical points of f. (c) Find any inflection points of f. (d) Evaluate f at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of f in the interval. (e) Graph f. f(x) = x3 3x2 (1 x 3)

For the functions in Problems 1619, do the following: (a) Find f and f . (b) Find the critical points of f. (c) Find any inflection points of f. (d) Evaluate f at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of f in the interval. (e) Graph f. f(x) = 2x3 9x2 + 12x + 1 (0.5 x 3)

For the functions in Problems 1619, do the following: (a) Find f and f . (b) Find the critical points of f. (c) Find any inflection points of f. (d) Evaluate f at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of f in the interval. (e) Graph f. f(x) = x3 3x2 9x +15 (5 x 4)

For the functions in Problems 1619, do the following: (a) Find f and f . (b) Find the critical points of f. (c) Find any inflection points of f. (d) Evaluate f at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of f in the interval. (e) Graph f. f(x) = x + sinx (0 x 2)

Find the value of x that maximizes y = 12+18x 5x2 and the corresponding value of y, by (a) Estimating the values from a graph of y. (b) Finding the values using calculus.

Find the value(s) of x that give critical points of y = ax2 + bx + c, where a, b, c are constants. Under what conditions on a, b, c is the critical value a maximum? A minimum?

A grapefruit is tossed straight up with an initial velocity of 50 ft/sec. The grapefruit is 5 feet above the ground when it is released. Its height, in feet, at time t seconds is given by y = 16t2 + 50t + 5. How high does it go before returning to the ground?

The sum of two nonnegative numbers is 100. What is the maximum value of the product of these two numbers?

The product of two positive numbers is 784. What is the minimum value of their sum?

The sum of three nonnegative numbers is 36, and one of the numbers is twice one of the other numbers. What is the maximum value of the product of these three numbers?

The perimeter of a rectangle is 64 cm. Find the lengths of the sides of the rectangle giving the maximum area.

In Problems 2732, find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. g(x) = 4x x2 5

In Problems 2732, find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. f(x) = x+ 1/x for x > 0

In Problems 2732, find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. g(t) = tet for t > 0

In Problems 2732, find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. f(x) = x ln x for x > 0

In Problems 2732, find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. f(t) = t 1 + t2

In Problems 2732, find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. f(t) = (sin2 t + 2) cos t

What value of w minimizes S if S5pw = 3qw26pq and p and q are positive constants?

The energy expended by a bird per day, E, depends on the time spent foraging for food per day, F hours. Foraging for a shorter time requires better territory, which then requires more energy for its defense.3 Find the foraging time that minimizes energy expenditure if E = 0.25F + 1.7 F2 .

A rectangular swimming pool is to be built with an area of 1800 square feet. The owner wants 5-foot-wide decks along either side and 10-foot-wide decks at the two ends. Find the dimensions of the smallest piece of property on which the pool can be built satisfying these conditions.

If you have 100 feet of fencing and want to enclose a rectangular area up against a long, straight wall, what is the largest area you can enclose?

An apple tree produces, on average, 400 kg of fruit each season. However, if more than 200 trees are planted per km2, crowding reduces the yield by 1 kg for each tree over 200. (a) Express the total yield, y, fromone square kilometer as a function of the number of trees on it. Graph this function. (b) How many trees should a farmer plant on each square kilometer to maximize yield?

On the west coast of Canada, crows eat whelks (a shellfish). To open the whelks, the crows drop them from the air onto a rock. If the shell does not smash the first time, the whelk is dropped again.4 The average number of drops, n, needed when the whelk is dropped from a height of x meters is approximated by n(x) = 1+ 27 x2 . (a) Give the total vertical distance the crow travels upward to open a whelk as a function of drop height, x. (b) Crows are observed to drop whelks from the height that minimizes the total vertical upward distance traveled per whelk. What is this height?

During a flu outbreak in a school of 763 children, the number of infected children, I, was expressed in terms of the number of susceptible (but still healthy) children, S, by the function5 I = 192ln$ S 762% S + 763. What is the maximum possible number of infected children?

The number of offspring in a population may not be a linear function of the number of adults. The Ricker curve, used to model fish populations, claims that y = axebx, where x is the number of adults, y is the number of offspring, and a and b are positive constants. (a) Find and classify all critical points of the Ricker curve. (b) Is there a global maximum? What does this imply about populations?

The oxygen supply, S, in the blood depends on the hematocrit, H, the percentage of red blood cells in the blood: S = aHebH for positive constants a, b. (a) What value of H maximizes the oxygen supply? What is the maximum oxygen supply? (b) How does increasing the value of the constants a and b change the maximum value of S?

The quantity of a drug in the bloodstream t hours after a tablet is swallowed is given, in mg, by q(t) = 20(et e2t). (a) How much of the drug is in the bloodstream at time t = 0? (b) When is the maximum quantity of drug in the bloodstream? What is that maximum? (c) In the long run, what happens to the quantity?

When birds lay eggs, they do so in clutches of several at a time. When the eggs hatch, each clutch gives rise to a brood of baby birds. We want to determine the clutch size which maximizes the number of birds surviving to adulthood per brood. If the clutch is small, there are few baby birds in the brood; if the clutch is large, there are so many baby birds to feed that most die of starvation. The number of surviving birds per brood as a function of clutch size is shown by the benefit curve in Figure 4.41.6 (a) Estimate the clutch size which maximizes the number of survivors per brood. (b) Suppose also that there is a biological cost to having a larger clutch: the female survival rate is reduced by large clutches. This cost is represented by the dotted line in Figure 4.41. If we take cost into account by assuming that the optimal clutch size in fact maximizes the vertical distance between the curves, what is the new optimal clutch size? 5 10 15 clutch size benefit or cost Cost: adult mortality Benefit: number of surviving young Figure 4.41

Let f(v) be the amount of energy consumed by a flying bird, measured in joules per second (a joule is a unit of energy), as a function of its speed v (in meters/sec). Let a(v) be the amount of energy consumed by the same bird, measured in joules per meter. (a) Suggest a reason in terms of the way birds fly for the shape of the graph of f(v) in Figure 4.42. (b) What is the relationship between f(v) and a(v)? (c) Where on the graph is a(v) a minimum? (d) Should the bird try to minimize f(v) or a(v) when it is flying? Why? f(v) v, speed (m/sec) energy (joules/sec) Figure 4.42

A persons blood pressure, p, in millimeters of mercury (mm Hg) is given, for t in seconds, by p = 100+20sin(2.5t). (a) What are the maximum and minimum values of blood pressure? (b) What is the time between successive maxima? (c) Show your answers on a graph of blood pressure against time.

A chemical reaction converts substance A to substance Y . At the start of the reaction, the quantity of A present is a grams. At time t seconds later, the quantity of Y present is y grams. The rate of the reaction, in grams/sec, is given by Rate = ky(a y), kis a positive constant. (a) For what values of y is the rate nonnegative? Graph the rate against y. (b) For what values of y is the rate a maximum?

In a chemical reaction, substance A combines with substance B to form substance Y . At the start of the reaction, the quantity of A present is a grams, and the quantity of B present is b grams. At time t seconds after the start of the reaction, the quantity of Y present is y grams. Assume a < b and y a. For certain types of reactions, the rate of the reaction, in grams/sec, is given by Rate = k(a y)(b y), kis a positive constant. (a) For what values of y is the rate nonnegative? Graph the rate against y. (b) Use your graph to find the value of y at which the rate of the reaction is fastest.