Suppose thatfis a function with the properties (i) fis differ-entiable everywhere, (ii)fix+y)=f(x)fly)for all valuesofxandy, (iii)f(0)=0, and (iv)f (0)=1.(a) Show thatf(0)=1. [Hint:Considerf(0+0).](b) Show thatf(x) >0 for all values ofx_[Hint: First showthatf(x)=0 for anyxby consideringf(x-x).](c) Use the definition of derivative (Definition 2.2.1) to showthatf"(x)=f(x)for all values ofx.2.Suppose thatfandgare functions each of which has theproperties (i)-(iv) in Exercise 1.(e) Show thaty=f(2x)satisfies the equationy'=2yintwo ways: using property (ii), and by directly applyingthe chain rule (Theorem 2.6.1).(b) Ifkis any constant, show thaty=f(kx)satisfies theequationy"=ky.(c) Find a value ofksuch thaty=f(x)g(x)satisfies theequationy'=ky.(d) Ifh=f/g, findh (x). Make a conjecture about therelationship betweenfandg.3.(e) Apply the product rule (Theorem 2.4.1) twice to showthat iffig, andhare differentiable functions, thenf g-his differentiable and(fig-h)=fig-h+fightfig-h'(b) Suppose thatf, g, h, andkare differentiable functions. Derive a formula for(fig.h-k)' A rectangular field is to be bounded by a fence on three sidesand by a straight stream on the fourth side. Find the dimen-sions of the field with maximum area that can be enclosedusing 1000 ft of fence.4. The boundary of a field is a right triangle with a straightstream along its hypotenuse and with fences along its othertwo sides. Find the dimensions of the field with maximumarea that can be enclosed using 1000 ft of fence.5.A rectangular plot of land is to be fenced in using two kindsof fencing. Two opposite sides will use heavy-duty fencing selling for $3 a foot, while the remaining two sides willuse standard fencing selling for $2 a foot. What are the di-mensions of the rectangular plot of greatest area that can befenced in at a cost of $600076.A rectangle is to be inscribed in a right triangle having sidesof length 6 in, & in, and 10 in. Find the dimensions ofthe rectangle with greatest area assuming the rectangle ispositioned A box with a square base is wider than it is tall. In order tosend the box through the U.S. mail, the width of the box andthe perimeter of one of the (nonsquare) sides of the box cansum to no more than 108 in. What is the maximum volumefor such a box?21.An open box is to be made froma3itbyBftrectangularpiece of sheet metal by cutting out squares of equal sizefrom the four corners and bending up the sides. Find themaximum volume that the box can have.22.A closed rectangular container with a square base is to havea volume of 2250 ins. The material for the top and bottomof the container will cost $2 per in:, and the material forthe sides will cost $3 per inc. Find the dimensions of thecontainer of least cost.23.A closed rectangular container with a square base is to havea volume of 2000 cma. It costs twice as much per squarecentimeter for the top and bottom as it does for the sides. Find the dimensions of the container of least cost
