Solve question 10 completely LTE8:38 AM OVOLTE e 4 28%192 Derivatives and Integralsp1, 6)6. (a) Let (xo, yo) be a point of the plane, and let L be the graph of thefunction f(x) = mx + b. Find the point F such that the distancefrom (x, yo) to (z, f(7)) is smallest. [Notice that minimizing thisdistance is the same as minimizing its square. This may simplifythe computations somewhat.](b) Also find z by noting that the line from (xo, yu) to (7, f(7)) is perpen-(0, a)a(x, 0)dicular to L.(0, -a)o(c) Find the distance from (x, yo) to L, i.e., the distance from (xo, yo) to(x, f(x)). (It will make the computations easier if you first assumethat 6 = 0; then apply the result to the graph of f(x) - mx and thepoint (xo, yo - b).] Compare with Problem 4-22.(d) Consider a straight line described by the equation Ax + By +C = 0(Problem 4-7). Show that the distance from (xo, yo) to this line is(Ax, + By, + C)/VAª + B³.The previous Problem suggests the following question: What is therelationship between the critical points of f and those of f?FIGURE 23Surface arca is the7.sum of these areas8. A straight line is drawn from the point (0, a) to the horizontal axis, andthen back to (1, 6), as in Figure 23. Prove that the total length is shortestwhen the angles a and ß are equal. (Naturally you must bring a functioninto the picture: express the length in terms of x, where (x, 0) is the pointon the horizontal axis. The dashed line in Figure 23 suggests an alterna-tive geometric proof; in either case the problem can be solvcd withoutactually finding the point (x, 0).)FIGURE 249.Prove that of all rectangles with given perimeter, the square has thegreatest area.10. Find, among all right circular cylinders of fixed volume V, the one withsmallest surface area (counting the areas of the faces at top and bottom,as in Figure 24).11. A right triangle with hypotenuse of length a is rotated about one of itslegs to generate a right circular cone. Find the greatest possible volumeof such a cone.FIGURE 2512. Two hallways, of widths a and b, meet at right angles (Figure 25). Whatis the greatest possible length of a ladder which can be carried hori-zontally around the corner?13. A garden is to be designed in the shape of a circular sector (Figure 26),with radius R and central angle 0. The garden is to have a fixed area A.For what value of R and 0 (in radians) will the length of the fencingaround the perimeter be minimized?Show that the sum of a number and its reciprocal is at least 2.15. Find the trapezoid of largest area that can be inscribed in a semicircle ofradius a, with one base lying along the diameter.R14.FIGURE 2611. Significance of the DerivativeA right angle is moved along thc diameter of a circle of radius eshown in Figure 27. What is the greatest possible length (A + B) incepted on it by the circle?17. Ecological Ed must cross a circular lake of radius 1 mile. He canacross at 2 mph or walk around at 4 mph, or he can row part waywalk the rest (Figure 28). What route should he take so as to16.B(i) see as much scenery as possible?(ii) cross as quickly as possible?18. The lower right-hand corner of a page is folded over so that itFIGURE 27touches the left edge of the paper, as in Figure 29. If the width ofpaper is a and the page is very long, show that the minimum lengtthe crease is 3V3a/4.19. Figure 30 chows the aranh of thá derinatiue cf f Find all local mavinrowwalk

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10. Find, among all right circular cylinders of fixed volume V, the one with smallest surface area (counting the areas of the faces at top and bottom, as in Figure 24).