- If f
^{′}= 0 , is it necessary that x is either a local minimum or local maximum? Hint: Consider f= x^{3}. - A continuous function f defined on is to be maximized. It was shown above in Theorem 7.6.2 that if the maximum value of f occurs at x ∈, and if f is differentiable there, then f
^{′}= 0 . However, this theorem does not say anything about the case where the maximum of f occurs at either a or b. Describe how to find the point ofwhere f achieves its maximum. Does f have a maximum? Explain. - Show that if the maximum value of a function f differentiable on occurs at the right endpoint, then for all h > 0,f
^{′}h ≥ 0. This is an example of a variational inequality. Describe what happens if the maximum occurs at the left end point and give a similar variational inequality. What is the situation for minima? - Find the maximum and minimum values and the values of x where these are
achieved for the function f= x +.
- A piece of wire of length L is to be cut in two pieces. One piece is bent into the shape of an equilateral triangle and the other piece is bent to form a square. How should the wire be cut to maximize the sum of the areas of the two shapes? How should the wire be bent to minimize the sum of the areas of the two shapes? Hint: Be sure to consider the case where all the wire is devoted to one of the shapes separately. This is a possible solution even though the derivative is not zero there.
- Lets find the point on the graph of y = which is closest to. One way to do it is to observe that a typical point on the graph is of the formand then to minimize the function f= x
^{2}+^{2}. Taking the derivative of f yields x +x^{3}and setting this equal to 0 leads to the solution, x = 0. Therefore, the point closest tois. Now lets do it another way. Lets use y =to write x^{2}= 4y. Now foron the graph, it follows it is of the form. Therefore, minimize f= 4 y +^{2}. Take the derivative to obtain 2 + 2y which requires y = −1. However, on this graph, y is never negative. What on earth is the problem? - Find the dimensions of the largest rectangle that can be inscribed in the ellipse,
+= 1 .
- A function f, is said to be odd if f= −fand a function is said to be even if f= f. Show that if f is even, then f
^{′}is odd and if f is odd, then f^{′}is even. Sketch the graph of a typical odd function and a typical even function. - Find the point on the curve, y = which is closest to.
- A street is 200 feet long and there are two lights located at the ends of the street.
One of the lights is times as bright as the other. Assuming the brightness of light from one of these street lights is proportional to the brightness of the light and the reciprocal of the square of the distance from the light, locate the darkest point on the street.
- Find the volume of the smallest right circular cone which can be circumscribed
about a sphere of radius 4 inches.
- Show that for r a rational number and y = x
^{r}, it must be the case that if this function is differentiable, then y^{′}= rx^{r−1}. - Let f be a continuous function defined on . Let ε > 0 be given. Show there exists a polynomial p such that for all x ∈,
This follows from the Weierstrass approximation theorem, Theorem 6.10.3. Now here is the interesting part. Show there exists a function g which is also continuous on

and for all x ∈,but g has no derivative at any point. Thus there are enough nowhere differentiable functions that any continuous function is uniformly close to one. Explain why every continuous function is the uniform limit of nowhere differentiable functions. Also explain why every nowhere differentiable continuous function is the uniform limit of polynomials. Hint: You should look at the construction of the nowhere differentiable function which is everywhere continuous, given above.

- Consider the following nested sequence of compact sets, {P
_{n}}. Let P_{1}=, P_{2}=∪, etc. To go from P_{n}to P_{n+1}, delete the open interval which is the middle third of each closed interval in P_{n}. Let P = ∩_{n=1}^{∞}P_{n}. By Problem 16 on Page 155, P≠∅. If you have not worked this exercise, now is the time to do it. Show the total length of intervals removed fromis equal to 1. If you feel ambitious also show there is a one to one onto mapping of [0 ,1] to P. The set P is called the Cantor set. Thus P has the same number of points in it asin the sense that there is a one to one and onto mapping from one to the other even though the length of the intervals removed equals 1. Hint: There are various ways of doing this last part but the most enlightenment is obtained by exploiting the construction of the Cantor set rather than some silly representation in terms of sums of powers of two and three. All you need to do is use the theorems in the chapter on set theory related to the Schroder Bernstein theorem and show there is an onto map from the Cantor set to. If you do this right it will provide a construction which is very useful to prove some even more surprising theorems which you may encounter later if you study compact metric spaces. The Cantor set is just a simple version of what is seen in some vegetables. Note in the following picture of a kind of broccoli, the spirals of points each of which is a spiral of points each of which is a spiral of points...

- ↑ Consider the sequence of functions defined in the following way. Let f
_{1}= x on [0,1]. To get from f_{n}to f_{n+1}, let f_{n+1}= f_{n}on all intervals where f_{n}is constant. If f_{n}is nonconstant on [a,b], let f_{n+1}(a) = f_{n}(a),f_{n+1}(b) = f_{n}(b),f_{n+1}is piecewise linear and equal to( f_{n}(a) + f_{n}(b)) on the middle third of [a,b]. Sketch a few of these and you will see the pattern. The process of modifying a nonconstant section of the graph of this function is illustrated in the following picture.Show {f

_{n}} converges uniformly on [0,1]. If f(x) = lim_{n→∞}f_{n}(x), show that f(0) = 0,f(1) = 1,f is continuous, and f^{′}(x) = 0 for all xP where P is the Cantor set of Problem 14. This function is called the Cantor function.It is a very important example to remember especially for those who like mathematical pathology. Note it has derivative equal to zero on all those intervals which were removed and whose total length was equal to 1 and yet it succeeds in climbing from 0 to 1. Isn’t this amazing? Hint: This isn’t too hard if you focus on getting a careful estimate on the difference between two successive functions in the list considering only a typical small interval in which the change takes place. The above picture should be helpful. - Let
Now let g

= x^{2}f. Find where g is continuous and differentiable if anywhere.

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